MikaStars39/nano-eval huggingface dataset:
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-2+2 \sqrt{3} i}$.

Final answer: The final answer is -2+2 \sqrt{3} i. I hope it is correct.

Subproblem 2: Rewrite $e^{3(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -8
Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{\cos (2 t)}{15}
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -2+2 \sqrt{3} i
Find a solution of $\ddot{x}+4 x=\cos (2 t)$, solving for $x(t)$, by using the ERF on a complex replacement. The ERF (Exponential Response Formula) states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{e^{r t}}{p(r)}$, as long as $\left.p (r\right) \neq 0$). The ERF with resonance assumes that $p(r)=0$ and states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{t e^{r t}}{p^{\prime}(r)}$, as long as $\left.p^{\prime} ( r\right) \neq 0$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{t}{4} \sin (2 t)
Given the ordinary differential equation $\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 1$ and $\dot{x}(0)=0$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{1}{2}(\exp{a*t} + \exp{-a*t})
Find the general solution of the differential equation $\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{e^{t}} {3}+c e^{-2 t}
Find a solution of $\ddot{x}+3 \dot{x}+2 x=t e^{-t}$ in the form $x(t)=u(t) e^{-t}$ for some function $u(t)$.  Use $C$ for an arbitrary constant, should it arise.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \left(\frac{t^{2}}{2}-t+C\right) e^{-t}
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -2
Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{e^{-2 t}}{15}
Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$. 

What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: m
Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$

Subproblem 0: Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$.


Solution: We can find this by brute force. If $x=A \sin (3 t)$, then $\ddot{x}=-9 A \sin (3 t)$, so $\ddot{x}+4 x=-5 A \sin (3 t)$. Therefore, when $A=\boxed{-0.2}, x_{p}(t)=-\sin (3 t) / 5$ is a solution of the given equation.

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

Subproblem 1: What is the general solution, in the form $f_0(t) + C_1f_1(t) + C_2f_2(t)$, where $C_1, C_2$ denote arbitrary constants?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -\sin (3 t) / 5+ C_{1} \sin (2 t)+C_{2} \cos (2 t)
What is the smallest possible positive $k$ such that all functions $x(t)=A \cos (\omega t-\phi)$---where $\phi$ is an odd multiple of $k$---satisfy $x(0)=0$? \\
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{\pi}{2}
Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\

What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: s^{2}+b s+1
Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$. 

Subproblem 0: What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.


Solution: We can write $e^{-t / 2} \cos (3 t)=\operatorname{Re} e^{(-1 / 2 \pm 3 i) t}$, so $p(s)=m s^{2}+b s+k$ has solutions $-\frac{1}{2} \pm 3 i$. This means $p(s)=m(s+1 / 2-3 i)(s+1 / 2+3 i)=m\left(s^{2}+s+\frac{37}{4}\right)$. Then $b=\boxed{m}$, 

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

Subproblem 1: What is $k$ in terms of $m$? Write $k$ as a constant times a function of $m$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{37}{4} m
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-2+2 \sqrt{3} i}$.

Final answer: The final answer is -2+2 \sqrt{3} i. I hope it is correct.

Subproblem 2: Rewrite $e^{3(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-8}$.

Final answer: The final answer is -8. I hope it is correct.

Subproblem 3: Rewrite $e^{4(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -8-8 \sqrt{3} i
Rewrite the function $\operatorname{Re} \frac{e^{i t}}{2+2 i}$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{\sqrt{2}}{4} \cos (t-\pi / 4)
Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\

Subproblem 0: What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$?


Solution: The characteristic polynomial is $p(s)=\boxed{s^{2}+b s+1}$.

Final answer: The final answer is s^{2}+b s+1. I hope it is correct.

Subproblem 1: For what value of $b$ does $\ddot{x}+b \dot{x}+x=0$ exhibit critical damping?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 2
Find the general (complex-valued) solution of the differential equation $\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: \frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}
Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\). 

Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final value of 1.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: -\tau \ln \Delta
Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\). 

Subproblem 0: Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final value of 1.


Solution: Rise and Settling Times.  We are given the first-order transfer function
\[
H(s)=\frac{1}{\tau s+1}
\]
The response to a unit step with zero initial conditions will be \(y(t)=1-e^{-t / \tau}\). To determine the amount of time it take \(y\) to settle to within \(\Delta\) of its final value, we want to find the time \(t_{s}\) such that \(y\left(t_{s}\right)=1-\Delta\). Thus, we obtain
\[
\begin{aligned}
&\Delta=e^{-t_{s} / \tau} \\
&t_{s}=\boxed{-\tau \ln \Delta}
\end{aligned}
\]

Final answer: The final answer is -\tau \ln \Delta. I hope it is correct.

Subproblem 1: Derive an expression for the \(10-90 \%\) rise time \(t_{r}\) in terms of $\tau$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 2.2 \tau
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{{}}.
Answer: \frac{1}{s+a}
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{{}}.
Answer: \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)
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{{}}.
Answer: \frac{1}{s+1}-\frac{e^{-1}}{s}-\frac{e^{-1} e^{-s}}{s+1}+\frac{e^{-1} e^{-s}}{s}
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{{}}.
Answer: 20
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{{}}.
Answer: \frac{1}{s}-\frac{1}{s} e^{-s T}
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{{}}.
Answer: \frac{1}{m} t
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{{}}.
Answer: \frac{s+\sigma}{(s+\sigma)^{2}+\omega_{d}^{2}}
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{{}}.
Answer: \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}
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{{}}.
Answer: m \frac{d v}{d t}+b v=f
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{{}}.
Answer: 1000
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{{}}.
Answer: 0.10
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{{}}.
Answer: (1 - 3e^{-2t} + 2e^{-5t}) u(t)
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{{}}.
Answer: b \delta(t)+1-e^{-a t}
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{{}}.
Answer: \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)
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{{}}.
Answer: I \frac{d \omega}{d t}+2 B \omega=T
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{{}}.
Answer: (t + \frac{1}{2} \sin{2t}) u(t)
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{{}}.
Answer: \frac{1}{L C s^{2}+R C s+1}
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{{}}.
Answer: 0.2
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{{}}.
Answer: 1e-8
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{{}}.
Answer: 100
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{{}}.
Answer: \frac{d^{2} v_{o}}{d t^{2}}+\frac{v_{o}}{L C}=\frac{v_{i}}{L C}
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{{}}.
Answer: m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t)
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{{}}.
Answer: 99.9
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{{}}.
Answer: 0.05
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{{}}.
Answer: 3e8
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{{}}.
Answer: 1e11
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{{}}.
Answer: \frac{1}{\sqrt{2}}c
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{{}}.
Answer: m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta
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{{}}.
Answer: \lambda_{\text {red}} / \sqrt{2}
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{{}}.
Answer: 10
How many down quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 5
How many up quarks does a tritium ($H^3$) nucleus contain?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 4
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{{}}.
Answer: 1
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{{}}.
Answer: 8
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{{}}.
Answer: 400000
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{{}}.
Answer: 560
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{{}}.
Answer: 671.02
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{{}}.
Answer: +2
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{{}}.
Answer: 1.4903e4
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{{}}.
Answer: 4.03188
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{{}}.
Answer: 566
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{{}}.
Answer: 4100
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{{}}.
Answer: 2.78
Calculate the "Bohr radius" in angstroms to 3 decimal places for ${He}^{+}$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 0.264
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

the frequency $(v)$ in Hz, to 4 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 4.4677
Electromagnetic radiation of frequency $3.091 \times 10^{14} \mathrm{~Hz}$ illuminates a crystal of germanium (Ge). Calculate the wavelength of photoemission in meters generated by this interaction. Germanium is an elemental semiconductor with a band gap, $E_{g}$, of $0.7 \mathrm{eV}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.77e-6
What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.3
Determine for hydrogen the velocity in m/s of an electron in an ${n}=4$ state. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 5.47e5
Preamble: A pure crystalline material (no impurities or dopants are present) appears red in transmitted light.

Subproblem 0: Is this material a conductor, semiconductor or insulator? Give the reasons for your answer.


Solution: If the material is pure (no impurity states present), then it must be classified as a \boxed{semiconductor} since it exhibits a finite "band gap" - i.e. to activate charge carriers, photons with energies in excess of "red" radiation are required.

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

Subproblem 1: What is the approximate band gap $\left(\mathrm{E}_{g}\right)$ for this material in eV? Please round your answer to 1 decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.9
Calculate the minimum potential $(V)$ in volts (to 1 decimal place) which must be applied to a free electron so that it has enough energy to excite, upon impact, the electron in a hydrogen atom from its ground state to a state of $n=5$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 13.1
Preamble: For light with a wavelength $(\lambda)$ of $408 \mathrm{~nm}$ determine:

Subproblem 0: the frequency in $s^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 3 decimal places. 


Solution: To solve this problem we must know the following relationships:
\[
\begin{aligned}
v \lambda &=c
\end{aligned}
\]
$v$ (frequency) $=\frac{c}{\lambda}=\frac{3 \times 10^{8} m / s}{408 \times 10^{-9} m}= \boxed{7.353e14} s^{-1}$

Final answer: The final answer is 7.353e14. I hope it is correct.

Subproblem 1: the wave number in $m^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places.


Solution: To solve this problem we must know the following relationships:
\[
\begin{aligned}
1 / \lambda=\bar{v} 
\end{aligned}
\]
$\bar{v}$ (wavenumber) $=\frac{1}{\lambda}=\frac{1}{408 \times 10^{-9} m}=\boxed{2.45e6} m^{-1}$

Final answer: The final answer is 2.45e6. I hope it is correct.

Subproblem 2: the wavelength in angstroms.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 4080
Preamble: Reference the information below to solve the following problems. 
$\begin{array}{llll}\text { Element } & \text { Ionization Potential }  & \text { Element } & \text { Ionization Potential } \\ {Na} & 5.14 & {Ca} & 6.11 \\ {Mg} & 7.64 & {Sc} & 6.54 \\ {Al} & 5.98 & {Ti} & 6.82 \\ {Si} & 8.15 & {~V} & 6.74 \\ {P} & 10.48 & {Cr} & 6.76 \\ {~S} & 10.36 & {Mn} & 7.43 \\ {Cl} & 13.01 & {Fe} & 7.9 \\ {Ar} & 15.75 & {Co} & 7.86 \\ & & {Ni} & 7.63 \\ & & {Cu} & 7.72\end{array}$

Subproblem 0: What is the first ionization energy (in J, to 3 decimal places) for Na?


Solution: The required data can be obtained by multiplying the ionization potentials (listed in the Periodic Table) with the electronic charge ( ${e}^{-}=1.6 \times 10^{-19}$ C).
\boxed{0.822} J.

Final answer: The final answer is 0.822. I hope it is correct.

Subproblem 1: What is the first ionization energy (in J, to 2 decimal places) for Mg?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.22
Light of wavelength $\lambda=4.28 \times 10^{-7} {~m}$ interacts with a "motionless" hydrogen atom. During this interaction it transfers all its energy to the orbiting electron of the hydrogen. What is the velocity in m/s of this electron after interaction? Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 2.19e6
Determine the minimum potential in V (to 2 decimal places) that must be applied to an $\alpha$-particle so that on interaction with a hydrogen atom, a ground state electron will be excited to $n$ $=6$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 6.62
Preamble: Reference the information below to solve the following problems. 
$\begin{array}{llll}\text { Element } & \text { Ionization Potential }  & \text { Element } & \text { Ionization Potential } \\ {Na} & 5.14 & {Ca} & 6.11 \\ {Mg} & 7.64 & {Sc} & 6.54 \\ {Al} & 5.98 & {Ti} & 6.82 \\ {Si} & 8.15 & {~V} & 6.74 \\ {P} & 10.48 & {Cr} & 6.76 \\ {~S} & 10.36 & {Mn} & 7.43 \\ {Cl} & 13.01 & {Fe} & 7.9 \\ {Ar} & 15.75 & {Co} & 7.86 \\ & & {Ni} & 7.63 \\ & & {Cu} & 7.72\end{array}$

What is the first ionization energy (in J, to 3 decimal places) for Na?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 0.822
Preamble: For "yellow radiation" (frequency, $v,=5.09 \times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:

Subproblem 0: the wavelength $(\lambda)$ in m. Please format your answer as $n \times 10^x$, where n is to 2 decimal places.


Solution: The equation relating $v$ and $\lambda$ is $c=v \lambda$ where $c$ is the speed of light $=3.00 \times 10^{8} \mathrm{~m}$.
\[
\lambda=\frac{c}{v}=\frac{3.00 \times 10^{8} m / s}{5.09 \times 10^{14} s^{-1}}=\boxed{5.89e-7} m
\]

Final answer: The final answer is 5.89e-7. 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 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.70e4
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?


Solution: \boxed{1}.

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

Subproblem 1: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{O}_{2}$ (in decimal form)?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 0.5
Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.

$\mathrm{NH}_{4} \mathrm{OH}$
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 35.06
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?


Solution: \boxed{1}.

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

Subproblem 1: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{O}_{2}$ (in decimal form)?


Solution: \boxed{0.5}. 

Final answer: The final answer is 0.5. I hope it is correct.

Subproblem 2: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}_{2}$ (in decimal form)?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1
Magnesium (Mg) has the following isotopic distribution:
\[
\begin{array}{ll}
24_{\mathrm{Mg}} & 23.985 \mathrm{amu} \text { at } 0.7870 \text { fractional abundance } \\
25_{\mathrm{Mg}} & 24.986 \mathrm{amu} \text { at } 0.1013 \text { fractional abundance } \\
26_{\mathrm{Mg}} & 25.983 \mathrm{amu} \text { at } 0.1117 \text { fractional abundance }
\end{array}
\]
What is the atomic weight of magnesium (Mg) (to 3 decimal places) according to these data?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 24.310
Preamble: Electrons are accelerated by a potential of 10 Volts.

Determine their velocity in m/s. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.87e6
Determine the frequency (in $s^{-1}$ of radiation capable of generating, in atomic hydrogen, free electrons which have a velocity of $1.3 \times 10^{6} {~ms}^{-1}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 4.45e15
In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1
Preamble: Electrons are accelerated by a potential of 10 Volts.

Subproblem 0: Determine their velocity in m/s. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places. 


Solution: The definition of an ${eV}$ is the energy gained by an electron when it is accelerated through a potential of $1 {~V}$, so an electron accelerated by a potential of $10 {~V}$ would have an energy of $10 {eV}$.\\
${E}=\frac{1}{2} m {v}^{2} \rightarrow {v}=\sqrt{2 {E} / {m}}$
\[
E=10 {eV}=1.60 \times 10^{-18} {~J}
\]
\[
\begin{aligned}
& {m}=\text { mass of electron }=9.11 \times 10^{-31} {~kg} \\
& v=\sqrt{\frac{2 \times 1.6 \times 10^{-18} {~J}}{9.11 \times 10^{-31} {~kg}}}= \boxed{1.87e6} {~m} / {s} 
\end{aligned}
\]

Final answer: The final answer is 1.87e6. I hope it is correct.

Subproblem 1: Determine their deBroglie wavelength $\left(\lambda_{p}\right)$ in m. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 3.89e-10
Preamble: In all likelihood, the Soviet Union and the United States together in the past exploded about ten hydrogen devices underground per year.

If each explosion converted about $10 \mathrm{~g}$ of matter into an equivalent amount of energy (a conservative estimate), how many $k J$ of energy were released per device? Please format your answer as $n \times 10^{x}$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 9e11
Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.

Subproblem 0: $\mathrm{NH}_{4} \mathrm{OH}$


Solution: $\mathrm{NH}_{4} \mathrm{OH}$ :
$5 \times 1.01=5.05(\mathrm{H})$
$1 \times 14.01=14.01(\mathrm{~N})$
$1 \times 16.00=16.00(\mathrm{O})$
$\mathrm{NH}_{4} \mathrm{OH}= \boxed{35.06}$ g/mole

Final answer: The final answer is 35.06. I hope it is correct.

Subproblem 1: $\mathrm{NaHCO}_{3}$


Solution: $\mathrm{NaHCO}_{3}: 3 \times 16.00=48.00(\mathrm{O})$
$1 \times 22.99=22.99(\mathrm{Na})$
$1 \times 1.01=1.01$ (H)
$1 \times 12.01=12.01$ (C)
$\mathrm{NaHCO}_{3}= \boxed{84.01}$ g/mole

Final answer: The final answer is 84.01. I hope it is correct.

Subproblem 2: $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}$
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 46.08
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?


Solution: \boxed{1}.

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

Subproblem 1: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{O}_{2}$ (in decimal form)?


Solution: \boxed{0.5}. 

Final answer: The final answer is 0.5. I hope it is correct.

Subproblem 2: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}_{2}$ (in decimal form)?


Solution: \boxed{1}.

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

Subproblem 3: If $32.0 \mathrm{~g}$ of oxygen react with $\mathrm{CO}$ to form carbon dioxide $\left(\mathrm{CO}_{2}\right)$, how much CO was consumed in this reaction (to 1 decimal place)?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 56.0
Preamble: For "yellow radiation" (frequency, $v,=5.09 \times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:

the wavelength $(\lambda)$ in m. Please format your answer as $n \times 10^x$, where n is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 5.89e-7
For a proton which has been subjected to an accelerating potential (V) of 15 Volts, determine its deBroglie wavelength in m. Please format your answer as $n \times 10^x$, where $n$ is to 1 decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 7.4e-12
Preamble: For light with a wavelength $(\lambda)$ of $408 \mathrm{~nm}$ determine:

the frequency in $s^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 3 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 7.353e14
Determine in units of eV (to 2 decimal places) the energy of a photon ( $h v)$ with the wavelength of $800$ nm.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.55
Determine for barium (Ba) the linear density of atoms along the $<110>$ directions, in atoms/m.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.39e9
A photon with a wavelength $(\lambda)$ of $3.091 \times 10^{-7} {~m}$ strikes an atom of hydrogen. Determine the velocity in m/s of an electron ejected from the excited state, $n=3$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 9.35e5
Preamble: For the element copper (Cu) determine:

the distance of second nearest neighbors (in meters). Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 3.61e-10
A line of the Lyman series of the spectrum of hydrogen has a wavelength of $9.50 \times 10^{-8} {~m}$. What was the "upper" quantum state $\left({n}_{{i}}\right)$ involved in the associated electron transition?
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 5
Determine the diffusivity $\mathrm{D}$ of lithium ( $\mathrm{Li}$ ) in silicon (Si) at $1200^{\circ} \mathrm{C}$, knowing that $D_{1100^{\circ} \mathrm{C}}=10^{-5} \mathrm{~cm}^{2} / \mathrm{s}$ and $\mathrm{D}_{695^{\circ} \mathrm{C}}=10^{-6} \mathrm{~cm}^{2} / \mathrm{s}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places, in $\mathrm{~cm}^2/\mathrm{sec}$.
Please reason step by step, and put your final answer within \boxed{{}}.
Answer: 1.45e-5