Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

$y(t)=e^{-a t}$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

$y(t)=e^{-a t}$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

$y(t)=e^{-a t}$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]

Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
therefore
\[
Y(s)=\frac{1}{b-a} \frac{1}{s+a}-\frac{1}{b-a} \frac{1}{s+b}
\]
By looking up the inverse Laplace Transform of \(\frac{1}{s+b}\), we find the total solution \(y(t)\)
\[
y(t)=\boxed{\frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)}
\]

Final answer: The final answer is \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right). I hope it is correct.

Subproblem 1: \[
Y(s)=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1}
\]
You may assume that $\zeta < 1$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

\omega_{n}^{2} e^{-\zeta \omega_{n} t} \cos \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)-\frac{\zeta \omega_{n}^{2}}{\sqrt{1-\zeta^{2}}} e^{-\zeta \omega_{n} t} \sin \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]

Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
therefore
\[
Y(s)=\frac{1}{b-a} \frac{1}{s+a}-\frac{1}{b-a} \frac{1}{s+b}
\]
By looking up the inverse Laplace Transform of \(\frac{1}{s+b}\), we find the total solution \(y(t)\)
\[
y(t)=\boxed{\frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)}
\]

Final answer: The final answer is \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right). I hope it is correct.

Subproblem 1: \[
Y(s)=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1}
\]
You may assume that $\zeta < 1$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

\omega_{n}^{2} e^{-\zeta \omega_{n} t} \cos \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)-\frac{\zeta \omega_{n}^{2}}{\sqrt{1-\zeta^{2}}} e^{-\zeta \omega_{n} t} \sin \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]

Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
therefore
\[
Y(s)=\frac{1}{b-a} \frac{1}{s+a}-\frac{1}{b-a} \frac{1}{s+b}
\]
By looking up the inverse Laplace Transform of \(\frac{1}{s+b}\), we find the total solution \(y(t)\)
\[
y(t)=\boxed{\frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)}
\]

Final answer: The final answer is \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right). I hope it is correct.

Subproblem 1: \[
Y(s)=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1}
\]
You may assume that $\zeta < 1$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

\omega_{n}^{2} e^{-\zeta \omega_{n} t} \cos \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)-\frac{\zeta \omega_{n}^{2}}{\sqrt{1-\zeta^{2}}} e^{-\zeta \omega_{n} t} \sin \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]

Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
therefore
\[
Y(s)=\frac{1}{b-a} \frac{1}{s+a}-\frac{1}{b-a} \frac{1}{s+b}
\]
By looking up the inverse Laplace Transform of \(\frac{1}{s+b}\), we find the total solution \(y(t)\)
\[
y(t)=\boxed{\frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)}
\]

Final answer: The final answer is \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right). I hope it is correct.

Subproblem 1: \[
Y(s)=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1}
\]
You may assume that $\zeta < 1$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

\omega_{n}^{2} e^{-\zeta \omega_{n} t} \cos \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)-\frac{\zeta \omega_{n}^{2}}{\sqrt{1-\zeta^{2}}} e^{-\zeta \omega_{n} t} \sin \left(\omega_{n} \sqrt{1-\zeta^{2}} t\right)


Prompt:
<|im_start|>user
A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

Subproblem 0: What is the time constant for this system?

Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t / \tau}$, meaning $\tau = \boxed{0.2}$.

Final answer: The final answer is 0.2. I hope it is correct.

Subproblem 1: If \(u=10\), what is the final or steady-state value for \(y(t)\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

Subproblem 0: What is the time constant for this system?

Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t / \tau}$, meaning $\tau = \boxed{0.2}$.

Final answer: The final answer is 0.2. I hope it is correct.

Subproblem 1: If \(u=10\), what is the final or steady-state value for \(y(t)\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

Subproblem 0: What is the time constant for this system?

Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t / \tau}$, meaning $\tau = \boxed{0.2}$.

Final answer: The final answer is 0.2. I hope it is correct.

Subproblem 1: If \(u=10\), what is the final or steady-state value for \(y(t)\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

Subproblem 0: What is the time constant for this system?

Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t / \tau}$, meaning $\tau = \boxed{0.2}$.

Final answer: The final answer is 0.2. I hope it is correct.

Subproblem 1: If \(u=10\), what is the final or steady-state value for \(y(t)\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end.  Assume zero initial velocity and position for the elements.
Recall that the unit step function \(u_{S}(t)\) is defined as \(u_{S}(t)=0 ; t<0\) and \(u_{S}(t)=1 ; t \geq 0\). We will also find it useful to introduce the unit impulse function \(\delta(t)\) which can be defined via
\[
u_{S}(t)=\int_{-\infty}^{t} \delta(\tau) d \tau
\]
This means that we can also view the unit impulse as the derivative of the unit step:
\[
\delta(t)=\frac{d u_{S}(t)}{d t}
\]

Solve for the resulting velocity of the mass.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end.  Assume zero initial velocity and position for the elements.
Recall that the unit step function \(u_{S}(t)\) is defined as \(u_{S}(t)=0 ; t<0\) and \(u_{S}(t)=1 ; t \geq 0\). We will also find it useful to introduce the unit impulse function \(\delta(t)\) which can be defined via
\[
u_{S}(t)=\int_{-\infty}^{t} \delta(\tau) d \tau
\]
This means that we can also view the unit impulse as the derivative of the unit step:
\[
\delta(t)=\frac{d u_{S}(t)}{d t}
\]

Solve for the resulting velocity of the mass.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end.  Assume zero initial velocity and position for the elements.
Recall that the unit step function \(u_{S}(t)\) is defined as \(u_{S}(t)=0 ; t<0\) and \(u_{S}(t)=1 ; t \geq 0\). We will also find it useful to introduce the unit impulse function \(\delta(t)\) which can be defined via
\[
u_{S}(t)=\int_{-\infty}^{t} \delta(\tau) d \tau
\]
This means that we can also view the unit impulse as the derivative of the unit step:
\[
\delta(t)=\frac{d u_{S}(t)}{d t}
\]

Solve for the resulting velocity of the mass.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end.  Assume zero initial velocity and position for the elements.
Recall that the unit step function \(u_{S}(t)\) is defined as \(u_{S}(t)=0 ; t<0\) and \(u_{S}(t)=1 ; t \geq 0\). We will also find it useful to introduce the unit impulse function \(\delta(t)\) which can be defined via
\[
u_{S}(t)=\int_{-\infty}^{t} \delta(\tau) d \tau
\]
This means that we can also view the unit impulse as the derivative of the unit step:
\[
\delta(t)=\frac{d u_{S}(t)}{d t}
\]

Solve for the resulting velocity of the mass.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$

Solution: \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\]

Final answer: The final answer is \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}. I hope it is correct.

Subproblem 2: $y(t)=e^{-\sigma t} \cos \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$

Solution: \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\]

Final answer: The final answer is \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}. I hope it is correct.

Subproblem 2: $y(t)=e^{-\sigma t} \cos \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$

Solution: \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\]

Final answer: The final answer is \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}. I hope it is correct.

Subproblem 2: $y(t)=e^{-\sigma t} \cos \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$

Solution: \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\]

Final answer: The final answer is \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}. I hope it is correct.

Subproblem 2: $y(t)=e^{-\sigma t} \cos \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$

Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Subproblem 0: Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.

Solution: The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\]

Final answer: The final answer is I \frac{d \omega}{d t}+2 B \omega=T. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
I &=0.001 \mathrm{~kg}-\mathrm{m}^{2} \\
B &=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(\omega_{s s}\), in radians per second, when the input is a constant torque of 10 Newton-meters?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Subproblem 0: Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.

Solution: The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\]

Final answer: The final answer is I \frac{d \omega}{d t}+2 B \omega=T. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
I &=0.001 \mathrm{~kg}-\mathrm{m}^{2} \\
B &=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(\omega_{s s}\), in radians per second, when the input is a constant torque of 10 Newton-meters?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Subproblem 0: Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.

Solution: The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\]

Final answer: The final answer is I \frac{d \omega}{d t}+2 B \omega=T. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
I &=0.001 \mathrm{~kg}-\mathrm{m}^{2} \\
B &=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(\omega_{s s}\), in radians per second, when the input is a constant torque of 10 Newton-meters?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Subproblem 0: Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.

Solution: The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\]

Final answer: The final answer is I \frac{d \omega}{d t}+2 B \omega=T. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
I &=0.001 \mathrm{~kg}-\mathrm{m}^{2} \\
B &=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(\omega_{s s}\), in radians per second, when the input is a constant torque of 10 Newton-meters?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.

Solution: The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\]

Final answer: The final answer is m \frac{d v}{d t}+b v=f. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
m &=1000 \mathrm{~kg} \\
b &=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(v_{s s}\) when the input is a constant force of 10 Newtons? Answer in meters per second.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.

Solution: The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\]

Final answer: The final answer is m \frac{d v}{d t}+b v=f. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
m &=1000 \mathrm{~kg} \\
b &=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(v_{s s}\) when the input is a constant force of 10 Newtons? Answer in meters per second.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.

Solution: The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\]

Final answer: The final answer is m \frac{d v}{d t}+b v=f. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
m &=1000 \mathrm{~kg} \\
b &=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(v_{s s}\) when the input is a constant force of 10 Newtons? Answer in meters per second.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).

Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.

Solution: The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\]

Final answer: The final answer is m \frac{d v}{d t}+b v=f. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
m &=1000 \mathrm{~kg} \\
b &=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(v_{s s}\) when the input is a constant force of 10 Newtons? Answer in meters per second.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

What is the time constant for this system?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

What is the time constant for this system?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

What is the time constant for this system?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

What is the time constant for this system?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Subproblem 0: Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).

Solution: Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\]

Final answer: The final answer is \frac{1}{L C s^{2}+R C s+1}. I hope it is correct.

Subproblem 1: Let \(L=0.01 \mathrm{H}\). Choose the value of $C$ such that \(\omega_{n}=10^{5}\) and \(\zeta=0.05\).  Give your answer in Farads.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Subproblem 0: Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).

Solution: Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\]

Final answer: The final answer is \frac{1}{L C s^{2}+R C s+1}. I hope it is correct.

Subproblem 1: Let \(L=0.01 \mathrm{H}\). Choose the value of $C$ such that \(\omega_{n}=10^{5}\) and \(\zeta=0.05\).  Give your answer in Farads.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Subproblem 0: Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).

Solution: Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\]

Final answer: The final answer is \frac{1}{L C s^{2}+R C s+1}. I hope it is correct.

Subproblem 1: Let \(L=0.01 \mathrm{H}\). Choose the value of $C$ such that \(\omega_{n}=10^{5}\) and \(\zeta=0.05\).  Give your answer in Farads.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Subproblem 0: Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).

Solution: Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\]

Final answer: The final answer is \frac{1}{L C s^{2}+R C s+1}. I hope it is correct.

Subproblem 1: Let \(L=0.01 \mathrm{H}\). Choose the value of $C$ such that \(\omega_{n}=10^{5}\) and \(\zeta=0.05\).  Give your answer in Farads.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$.  We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.

Write the governing differential equation for this circuit.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$.  We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.

Write the governing differential equation for this circuit.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$.  We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.

Write the governing differential equation for this circuit.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$.  We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.

Write the governing differential equation for this circuit.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane.  Let the rod (of length $l$) make an angle of $\theta$ with the vertical.  Gravity ($mg$) acts directly downward, the system input is a horizontal external force $f(t)$, and the system output is the angle $\theta(t)$.
Note: Do NOT make the small-angle approximation in your equation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t)


Prompt:
<|im_start|>user
Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane.  Let the rod (of length $l$) make an angle of $\theta$ with the vertical.  Gravity ($mg$) acts directly downward, the system input is a horizontal external force $f(t)$, and the system output is the angle $\theta(t)$.
Note: Do NOT make the small-angle approximation in your equation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t)


Prompt:
<|im_start|>user
Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane.  Let the rod (of length $l$) make an angle of $\theta$ with the vertical.  Gravity ($mg$) acts directly downward, the system input is a horizontal external force $f(t)$, and the system output is the angle $\theta(t)$.
Note: Do NOT make the small-angle approximation in your equation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t)


Prompt:
<|im_start|>user
Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane.  Let the rod (of length $l$) make an angle of $\theta$ with the vertical.  Gravity ($mg$) acts directly downward, the system input is a horizontal external force $f(t)$, and the system output is the angle $\theta(t)$.
Note: Do NOT make the small-angle approximation in your equation.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t)


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?

Solution: $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$

Final answer: The final answer is 0.05. I hope it is correct.

Subproblem 2: What is the value of the damped natural frequency \(\omega_{d}\) in radians per second? Give your answer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?

Solution: $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$

Final answer: The final answer is 0.05. I hope it is correct.

Subproblem 2: What is the value of the damped natural frequency \(\omega_{d}\) in radians per second? Give your answer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?

Solution: $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$

Final answer: The final answer is 0.05. I hope it is correct.

Subproblem 2: What is the value of the damped natural frequency \(\omega_{d}\) in radians per second? Give your answer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?

Solution: $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$

Final answer: The final answer is 0.05. I hope it is correct.

Subproblem 2: What is the value of the damped natural frequency \(\omega_{d}\) in radians per second? Give your answer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?

Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared with the proton rest mass $m_{p}$.

In the lab frame, the two protons have the same energy $\gamma m_{p}$ and impact angle $\theta$, and the electron is at rest. Calculate the energy $E_{\nu}$ of the neutrino in the rest frame of the deuteron in terms of $\theta, m_{p}$ and $\gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta


Prompt:
<|im_start|>user
Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared with the proton rest mass $m_{p}$.

In the lab frame, the two protons have the same energy $\gamma m_{p}$ and impact angle $\theta$, and the electron is at rest. Calculate the energy $E_{\nu}$ of the neutrino in the rest frame of the deuteron in terms of $\theta, m_{p}$ and $\gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta


Prompt:
<|im_start|>user
Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared with the proton rest mass $m_{p}$.

In the lab frame, the two protons have the same energy $\gamma m_{p}$ and impact angle $\theta$, and the electron is at rest. Calculate the energy $E_{\nu}$ of the neutrino in the rest frame of the deuteron in terms of $\theta, m_{p}$ and $\gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta


Prompt:
<|im_start|>user
Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared with the proton rest mass $m_{p}$.

In the lab frame, the two protons have the same energy $\gamma m_{p}$ and impact angle $\theta$, and the electron is at rest. Calculate the energy $E_{\nu}$ of the neutrino in the rest frame of the deuteron in terms of $\theta, m_{p}$ and $\gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.

Solution: Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$.

Final answer: The final answer is \frac{1}{\sqrt{2}}c. I hope it is correct.

Subproblem 1: A player standing on third base is wearing red socks emitting light of wavelength $\lambda_{\text {red}}$. What wavelength does Manny see in terms of $\lambda_{\text {red}}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.

Solution: Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$.

Final answer: The final answer is \frac{1}{\sqrt{2}}c. I hope it is correct.

Subproblem 1: A player standing on third base is wearing red socks emitting light of wavelength $\lambda_{\text {red}}$. What wavelength does Manny see in terms of $\lambda_{\text {red}}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.

Solution: Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$.

Final answer: The final answer is \frac{1}{\sqrt{2}}c. I hope it is correct.

Subproblem 1: A player standing on third base is wearing red socks emitting light of wavelength $\lambda_{\text {red}}$. What wavelength does Manny see in terms of $\lambda_{\text {red}}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.

Solution: Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$.

Final answer: The final answer is \frac{1}{\sqrt{2}}c. I hope it is correct.

Subproblem 1: A player standing on third base is wearing red socks emitting light of wavelength $\lambda_{\text {red}}$. What wavelength does Manny see in terms of $\lambda_{\text {red}}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many down quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many down quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many down quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many down quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many up quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many up quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many up quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
How many up quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Age of our universe when most He nuclei were formed in minutes:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Age of our universe when most He nuclei were formed in minutes:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Age of our universe when most He nuclei were formed in minutes:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Age of our universe when most He nuclei were formed in minutes:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)

Solution: \boxed{1e11}.

Final answer: The final answer is 1e11. I hope it is correct.

Subproblem 4: Light travel time to closest star (Sun!:) in minutes. (Please format your answer as an integer.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)

Solution: \boxed{1e11}.

Final answer: The final answer is 1e11. I hope it is correct.

Subproblem 4: Light travel time to closest star (Sun!:) in minutes. (Please format your answer as an integer.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)

Solution: \boxed{1e11}.

Final answer: The final answer is 1e11. I hope it is correct.

Subproblem 4: Light travel time to closest star (Sun!:) in minutes. (Please format your answer as an integer.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:

Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:

Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)

Solution: \boxed{1e11}.

Final answer: The final answer is 1e11. I hope it is correct.

Subproblem 4: Light travel time to closest star (Sun!:) in minutes. (Please format your answer as an integer.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.

Subproblem 0: Age of our universe when most He nuclei were formed in minutes:

Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of sufficient energy to eject electrons from a potassium photodiode surface?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of sufficient energy to eject electrons from a potassium photodiode surface?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of sufficient energy to eject electrons from a potassium photodiode surface?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of sufficient energy to eject electrons from a potassium photodiode surface?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.

Solution: $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$

Final answer: The final answer is 1.4903e4. I hope it is correct.

Subproblem 2: the wavelength $(\lambda)$ in nm, to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.

Solution: $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$

Final answer: The final answer is 1.4903e4. I hope it is correct.

Subproblem 2: the wavelength $(\lambda)$ in nm, to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.

Solution: $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$

Final answer: The final answer is 1.4903e4. I hope it is correct.

Subproblem 2: the wavelength $(\lambda)$ in nm, to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.

Solution: $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$

Final answer: The final answer is 1.4903e4. I hope it is correct.

Subproblem 2: the wavelength $(\lambda)$ in nm, to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.

Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\]

Final answer: The final answer is 566. I hope it is correct.

Subproblem 2: What is the minimum wavelength in m (to 2 decimal places) for broadcasts on the FM band?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\]

Final answer: The final answer is 566. I hope it is correct.

Subproblem 2: What is the minimum wavelength in m (to 2 decimal places) for broadcasts on the FM band?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\]

Final answer: The final answer is 566. I hope it is correct.

Subproblem 2: What is the minimum wavelength in m (to 2 decimal places) for broadcasts on the FM band?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Determine the following values from a standard radio dial.

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.

Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\]

Final answer: The final answer is 566. I hope it is correct.

Subproblem 2: What is the minimum wavelength in m (to 2 decimal places) for broadcasts on the FM band?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Calculate the "Bohr radius" in angstroms to 3 decimal places for ${He}^{+}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant