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For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$. | \frac{1}{4-\omega^{2}} | MINERVA_MATH | test |
Find a solution to $\dot{x}+2 x=\cos (2 t)$ in the form $k_0\left[f(k_1t) + g(k_2t)\right]$, where $f, g$ are trigonometric functions. Do not include homogeneous solutions to this ODE in your solution. | \frac{\cos (2 t)+\sin (2 t)}{4} | MINERVA_MATH | test |
Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$
Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$. | -0.2 | MINERVA_MATH | test |
Find the general solution of the differential equation $y^{\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \dot{x}+u p x=\frac{d}{d t}(u x)$ is an integrating factor. Additionally, note that a general solution to a differential equation has the form $x=x_... | x / 2-1 / 4+c e^{-2 x} | MINERVA_MATH | test |
Subproblem 0: Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.
Solution: The characteristic polynomial of the homogeneous equation is given by $p(s)=$ $s^{4}-1$. Since $p(-2)=15 \neq 0$, the exponential response formula gives the solution $\frac{e^{-2 t}}{p(-2)}=\boxed{\frac{e^{-2 t}}{15}}$... | \frac{e^{-2 t}}{15}+C_{1} e^{t}+C_{2} e^{-t}+ C_{3} \cos (t)+C_{4} \sin (t) | MINERVA_MATH | test |
Preamble: Consider the differential equation $\ddot{x}+\omega^{2} x=0$. \\
A differential equation $m \ddot{x}+b \dot{x}+k x=0$ (where $m, b$, and $k$ are real constants, and $m \neq 0$ ) has corresponding characteristic polynomial $p(s)=m s^{2}+b s+k$.\\
What is the characteristic polynomial $p(s)$ of $\ddot{x}+\omeg... | s^{2}+\omega^{2} | MINERVA_MATH | test |
Rewrite the function $\cos (\pi t)-\sqrt{3} \sin (\pi t)$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. | 2 \cos (\pi t+\pi / 3) | MINERVA_MATH | test |
Preamble: The following subproblems refer to the damped sinusoid $x(t)=A e^{-a t} \cos (\omega t)$.
What is the spacing between successive maxima of $x(t)$? Assume that $\omega \neq 0$. | 2 \pi / \omega | MINERVA_MATH | test |
Preamble: The following subproblems refer to a spring/mass/dashpot system driven through the spring modeled by the equation $m \ddot{x}+b \dot{x}+k x=k y$. Here $x$ measures the position of the mass, $y$ measures the position of the other end of the spring, and $x=y$ when the spring is relaxed.
In this system, regard ... | \frac{4 A}{3+3 i} e^{i t} | MINERVA_MATH | test |
Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circuit and by $V(t)$ the voltage increase across the voltage source, at time $t$. Denote by $R$ the resistance of the resistor and $C$ the capacita... | I(0) e^{-\frac{t}{R C}} | MINERVA_MATH | test |
Subproblem 0: 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.
Solution: Using integrating factors, we get $e^{2 t} z=e^{(2+2 i) t} /(2+2 i)+C$, or $z=\boxed{\frac{e^{2 i t}}{(2+2 i)}+C e^{... | \frac{e^{2 i t}}{(2+2 i)} | MINERVA_MATH | test |
Preamble: The following subproblems consider a second order mass/spring/dashpot system driven by a force $F_{\text {ext }}$ acting directly on the mass: $m \ddot{x}+b \dot{x}+k x=F_{\text {ext }}$. So the input signal is $F_{\text {ext }}$ and the system response is $x$. We're interested in sinusoidal input signal, $F_... | \frac{2-\omega^{2}-\omega i / 4}{\omega^{4}-\frac{63}{16} \omega^{2}+4} | MINERVA_MATH | test |
Preamble: The following subproblems refer to the following "mixing problem": A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \mathrm{gm} /$ liter is added at the rate of $r$ liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets so... | x^{\prime}+\frac{r}{V} x-r c=0 | MINERVA_MATH | test |
Find the polynomial solution of $\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$. | -t^2 - t - 3 | MINERVA_MATH | test |
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... | -8 | MINERVA_MATH | test |
Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$. | \frac{\cos (2 t)}{15} | MINERVA_MATH | test |
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... | -2+2 \sqrt{3} i | MINERVA_MATH | test |
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... | \frac{t}{4} \sin (2 t) | MINERVA_MATH | test |
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 $\dot{x}(0)=0$. | \frac{1}{2}(\exp{a*t} + \exp{-a*t}) | MINERVA_MATH | test |
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. | \frac{e^{t}} {3}+c e^{-2 t} | MINERVA_MATH | test |
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. | \left(\frac{t^{2}}{2}-t+C\right) e^{-t} | MINERVA_MATH | test |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$? | -2 | MINERVA_MATH | test |
Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$. | \frac{e^{-2 t}}{15} | MINERVA_MATH | test |
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 $m$. | m | MINERVA_MATH | test |
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)... | -\sin (3 t) / 5+ C_{1} \sin (2 t)+C_{2} \cos (2 t) | MINERVA_MATH | test |
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 $x(0)=0$? \\ | \frac{\pi}{2} | MINERVA_MATH | test |
Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\
What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$? | s^{2}+b s+1 | MINERVA_MATH | test |
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} ... | \frac{37}{4} m | MINERVA_MATH | test |
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... | -8-8 \sqrt{3} i | MINERVA_MATH | test |
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 $b$. | \frac{\sqrt{2}}{4} \cos (t-\pi / 4) | MINERVA_MATH | test |
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 i... | 2 | MINERVA_MATH | test |
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. | \frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t} | MINERVA_MATH | test |
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. | -\tau \ln \Delta | MINERVA_MATH | test |
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 ... | 2.2 \tau | MINERVA_MATH | test |
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :
$y(t)=e^{-a t}$ | \frac{1}{s+a} | MINERVA_MATH | test |
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... | \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) | MINERVA_MATH | test |
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. | \frac{1}{s+1}-\frac{e^{-1}}{s}-\frac{e^{-1} e^{-s}}{s+1}+\frac{e^{-1} e^{-s}}{s} | MINERVA_MATH | test |
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 ... | 20 | MINERVA_MATH | test |
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. | \frac{1}{s}-\frac{1}{s} e^{-s T} | MINERVA_MATH | test |
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 an... | \frac{1}{m} t | MINERVA_MATH | test |
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 correc... | \frac{s+\sigma}{(s+\sigma)^{2}+\omega_{d}^{2}} | MINERVA_MATH | test |
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 correc... | \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}} | MINERVA_MATH | test |
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. | m \frac{d v}{d t}+b v=f | MINERVA_MATH | test |
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 th... | 1000 | MINERVA_MATH | test |
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 \(f\)... | 0.10 | MINERVA_MATH | test |
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. | (1 - 3e^{-2t} + 2e^{-5t}) u(t) | MINERVA_MATH | test |
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 \(x(t), t>0\). | b \delta(t)+1-e^{-a t} | MINERVA_MATH | test |
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :
\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\] | \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right) | MINERVA_MATH | test |
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 velo... | I \frac{d \omega}{d t}+2 B \omega=T | MINERVA_MATH | test |
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. | (t + \frac{1}{2} \sin{2t}) u(t) | MINERVA_MATH | test |
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 \(V_{o}(s) / V_{i}(s)\). | \frac{1}{L C s^{2}+R C s+1} | MINERVA_MATH | test |
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? | 0.2 | MINERVA_MATH | test |
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... | 1e-8 | MINERVA_MATH | test |
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? | 100 | MINERVA_MATH | test |
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... | \frac{d^{2} v_{o}}{d t^{2}}+\frac{v_{o}}{L C}=\frac{v_{i}}{L C} | MINERVA_MATH | test |
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 ... | m l \ddot{\theta}(t)-m g \sin \theta(t)=f(t) \cos \theta(t) | MINERVA_MATH | test |
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 answe... | 99.9 | MINERVA_MATH | test |
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 answe... | 0.05 | MINERVA_MATH | test |
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. | 3e8 | MINERVA_MATH | test |
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 hydrog... | 1e11 | MINERVA_MATH | test |
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 tha... | \frac{1}{\sqrt{2}}c | MINERVA_MATH | test |
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 w... | m_{p} c^{2}\left(\gamma^{2}-1\right) \sin ^{2} \theta | MINERVA_MATH | test |
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 runnin... | \lambda_{\text {red}} / \sqrt{2} | MINERVA_MATH | test |
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 hydrog... | 10 | MINERVA_MATH | test |
How many down quarks does a tritium ($H^3$) nucleus contain? | 5 | MINERVA_MATH | test |
How many up quarks does a tritium ($H^3$) nucleus contain? | 4 | MINERVA_MATH | test |
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: | 1 | MINERVA_MATH | test |
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 hydrog... | 8 | MINERVA_MATH | test |
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 hydrog... | 400000 | MINERVA_MATH | test |
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 su... | 560 | MINERVA_MATH | test |
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 \... | 671.02 | MINERVA_MATH | test |
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n. | +2 | MINERVA_MATH | test |
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 \... | 1.4903e4 | MINERVA_MATH | test |
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents. | 4.03188 | MINERVA_MATH | test |
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_{\... | 566 | MINERVA_MATH | test |
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$. | 4100 | MINERVA_MATH | test |
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_{\... | 2.78 | MINERVA_MATH | test |
Calculate the "Bohr radius" in angstroms to 3 decimal places for ${He}^{+}$. | 0.264 | MINERVA_MATH | test |
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. | 4.4677 | MINERVA_MATH | test |
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 \... | 1.77e-6 | MINERVA_MATH | test |
What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom? | 1.3 | MINERVA_MATH | test |
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. | 5.47e5 | MINERVA_MATH | test |
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 \box... | 1.9 | MINERVA_MATH | test |
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 $n=5$. | 13.1 | MINERVA_MATH | test |
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{... | 4080 | MINERVA_MATH | test |
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 \\ ... | 1.22 | MINERVA_MATH | test |
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... | 2.19e6 | MINERVA_MATH | test |
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 $n$ $=6$. | 6.62 | MINERVA_MATH | test |
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 \\ ... | 0.822 | MINERVA_MATH | test |
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 ... | 1.70e4 | MINERVA_MATH | test |
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 $\ma... | 0.5 | MINERVA_MATH | test |
Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.
$\mathrm{NH}_{4} \mathrm{OH}$ | 35.06 | MINERVA_MATH | test |
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 $\ma... | 1 | MINERVA_MATH | test |
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 } 0.111... | 24.310 | MINERVA_MATH | test |
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. | 1.87e6 | MINERVA_MATH | test |
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. | 4.45e15 | MINERVA_MATH | test |
In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$? | 1 | MINERVA_MATH | test |
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 {~... | 3.89e-10 | MINERVA_MATH | test |
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 devic... | 9e11 | MINERVA_MATH | test |
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}_{... | 46.08 | MINERVA_MATH | test |
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