Dataset Viewer
question
stringlengths 25
853
| final_answer
stringlengths 9
103
|
|---|---|
An archer fires an arrow from the ground so that it passes through two hoops, which are both a height $h$ above the ground. The arrow passes through the first hoop one second after the arrow is launched, and through the second hoop another second later. What is the value of $h$?
(A) $5 m$.
(B) $10 m$.
(C) $12 m$.
(D) $15 m$.
(E) There is not enough information to decide.
|
\boxed{B}
|
An amusement park ride consists of a circular, horizontal room. A rider leans against its frictionless outer walls,which are angled back at $30^{\circ}$ with respect to the vertical, so that the rider's center of mass is $5.0 m$ from the center of the room. When the room begins to spin about its center, at what angular velocity will the rider's feet first lift off the floor?
(A) $1.9 \mathrm{rad}/\mathrm{s}$.
(B) $2.3 \mathrm{rad}/\mathrm{s}$.
(C) $3.5 \mathrm{rad}/\mathrm{s}$.
(D) $4.0 \mathrm{rad}/\mathrm{s}$.
(E) $5.6 \mathrm{rad}/\mathrm{s}$.
|
\boxed{A}
|
When a car's brakes are fully engaged, it takes $100 m$ to stop on a dry road, which has coefficient of kinetic friction $\mu_{k} = 0.8$ with the tires. Now suppose only the first $50 m$ of the road is dry, and the rest is covered with ice, with $\mu_{k} = 0.2$. What total distance does the car need to stop?
(A) $150 m$.
(B) $200 m$.
(C) $250 m$.
(D) $400 m$.
(E) $850 m$.
|
\boxed{C}
|
Two hemispherical shells can be pressed together to form a airtight sphere of radius $40 cm$. Suppose the shells are pressed together at a high altitude, where the air pressure is half its value at sea level. The sphere is then returned to sea level, where the air pressure is $10^{5}\mathrm{Pa}$. What force $F$, applied directly outward to each hemisphere, is required to pull them apart?
(A) $25,000 N$.
(B) $50,000 N$.
(C) $100,000 N$.
(D) $200,000 N$.
(E) $400,000 N$.
|
\boxed{A}
|
The viscous force between two plates of area $A$, with relative speed $v$ and separation $d$, is $F = {\eta Av}/d$, where $\eta$ is the viscosity. In fluid mechanics,the Ohnesorge number is a dimensionless number proportional to $\eta$ which characterizes the importance of viscous forces,in a drop of fluid of density $\rho$, surface tension $\gamma$, and length scale $\ell$. Which of the following could be the definition of the Ohnesorge number?
(A) $\frac{\eta \ell}{\sqrt{\rho \gamma}}$.
(B) $\eta \ell \sqrt{\frac{\rho}{\gamma}}$.
(C) $\eta \sqrt{\frac{\rho}{\gamma \ell}}$.
(D) $\eta \sqrt{\frac{\rho \ell}{\gamma}}$.
(E) $\frac{\eta}{\sqrt{\rho \gamma \ell}}$.
|
\boxed{E}
|
A syringe is filled with water of density $\rho$ and negligible viscosity. Its body is a cylinder of cross-sectional area $A_1$, which gradually tapers into a needle with cross-sectional area $A_2 \ll A_1$. The syringe is held in place and its end is slowly pushed inward by a force $F$, so that it moves with constant speed $v$. Water shoots straight out of the needle's tip. What is the approximate value of $F$?
(A) $\rho {v}^2{A}_{1}$.
(B) $\frac{\rho {v}^2{A}_{1}^2}{2{A}_{2}}$.
(C) $\frac{\rho {v}^2{A}_{1}^2}{{A}_{2}}$.
(D) $\frac{\rho {v}^2{A}_{1}^{3}}{2{A}_{2}^2}$.
(E) $\frac{\rho {v}^2{A}_{1}^{3}}{{A}_{2}^2}$
|
\boxed{D}
|
A spherical shell is made from a thin sheet of material with a mass per area of $\sigma$. Consider two points, $P_1$ and $P_2$, which are close to each other, but just inside and outside the sphere, respectively. If the accelerations due to gravity at these points are $g_1$ and $g_2$, respectively, what is the value of $|g_1 - g_2|$?
(A) $\pi G\sigma$.
(B) $4\pi G\sigma /3$.
(C) $2\pi G\sigma$.
(D) $4\pi G\sigma$.
(E) $8\pi G\sigma$.
|
\boxed{D}
|
Collisions between ping pong balls and paddles are not perfectly elastic. Suppose that if a player holds a paddle still and drops a ball on top of it from any height $h$, it will bounce back up to height $h/2$. To keep the ball bouncing steadily, the player moves the paddle up and down, so that it is moving upward with speed $1.0 m/s$ whenever the ball hits it. What is the height to which the ball is bouncing?
(A) $0.21 m$.
(B) $0.45 m$.
(C) $1.0 m$.
(D) $1.7 m$.
(E) There is not enough information to determine the height.
|
\boxed{D}
|
When a projectile falls through a fluid, it experiences a drag force proportional to the product of its cross-sectional area, the fluid density $\rho_{f}$, and the square of its speed. Suppose a sphere of density $\rho_{s} \gg \rho_{f}$ of radius $R$ is dropped in the fluid from rest. When the projectile has reached half of its terminal velocity, which of the following is its displacement proportional to?
(A) $R\sqrt{\rho_{s}/\rho_{f}}$.
(B) $R \rho_{s}/\rho_{f}$.
(C) $R (\rho_{s}/\rho_{f})^{3/2}$.
(D) $R (\rho_{s}/\rho_{f})^2$.
(E) $R (\rho_{s}/\rho_{f})^{3}$
|
\boxed{B}
|
Determine the electron's velocity $v$ in a circular orbit of radius $r$.
|
\boxed{$v = \sqrt{\frac{e^2}{4 \pi \varepsilon_0 m_e r}}$}
|
(1) Show that the radius of each orbit is given by $r_n = n^2 r_1$, where $r_1$ is called the Bohr radius. (2) Express $r_1$ in terms of $\alpha$, $m_e$, $c$ and $\hbar$ and (3) calculate its numerical value in $m$ with 3 digits. (4) Express $v_1$, the velocity on the orbit of radius $r_1$, in terms of $\alpha$ and $c$.
|
\boxed{$r_n = \frac{\hbar^2 n^2}{\alpha m_e c}$}
|
(1) Determine the electron's mechanical energy $E_n$ on an orbit of radius $r_n$ in terms of $e$, $\varepsilon_0$, $r_1$ and $n$.
(2) Determine $E_1$ in the ground state in terms of $\alpha$, $m_e$ and $c$.
(3)Compute the numerical value of $E_1$ in eV.
|
\boxed{$E_n = -\frac{e^2}{8 \pi \varepsilon_0 n^2 r_1}$}
|
Determine the magnitude $B_1$ of $\vec{B}_1$ at the position of the electron in terms of $\mu_0$, $e$, $\alpha$, $c$ and $r_1$.
|
\boxed{$B_1 = \frac{\mu_0 e \alpha c}{4 \pi r_1^2}$}
|
(1) Express $\Delta E_F$ as a function of $\alpha$ and $E_1$.
(2) Express the wavelength $\lambda_{HF}$ of a photon emitted during a transition between the two states of the hyperfine structure and (3) give its numerical value in $cm$ with two digits.
|
\boxed{$\lambda_{\text{HF}} = -\frac{hc}{3.72 \cdot \frac{m_e}{m_p} 4 \alpha^2 E_1}$}
|
In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\frac{d \varphi}{d r}$.
|
\boxed{$v_c = \sqrt{r \frac{d \varphi}{d r}}$}
|
(1) Find the equation of motion on $z$ for the vertical motion of a point mass $m$ in such a potential, assuming $r$ is constant.
(2) Show that, if $r < r_0$, the galactic plane is a stable equilibrium state by giving the angular frequency $\omega_0$ of small oscillations around it.
|
\boxed{$\ddot{z} = \frac{2z}{z_{0}^{2}} \varphi_{0} \ln(\frac{r}{r_{0}}) \exp[-(\frac{z}{z_{0}})^{2}]$}
|
(1) Identify the regime, either $r \gg r_m$ or $r \ll r_m$, in which the model of Eq. 1 recovers a potential of the form $\varphi_G(r, 0)$ with a suitable definition of $\varphi_0$.
(2) Write down the definition of $\varphi_0$.
(3) Under this condition $v_c(r)$ no longer depends on $r$. Express it in terms of $\varphi_0$.
|
\boxed{$\varphi_{0} = +4 \pi C_{m} G$}
|
For a simple pendulum clock, would thermal expansion make it go faster or slower?
(A) Faster.
(B) Slower.
|
\boxed{B}
|
Find the expression of the duration $\tau$ of the transform-limited pulse after the propagation distance of $L$, in terms of $\beta_2, \tau_p$, and $L$, where $\beta_{2}$ is the second-order propagation factor from expansion about the center frequency $\omega_{0}$. (You may ignore effect from third-order and above)
|
\boxed{$\tau = \tau_{p} \sqrt{1 + \frac{16 \beta_{2}^{2} L^{2}}{\tau_{p}^{4}}}$}
|
One critical step for high-harmonic generation (HHG) is to drive the bound charge of molecules in the non-perturbative regime to initiate the ionization dynamics. If we take hydrogen molecules for HHG, please estimate the peak electric field strength required to initiate the process (expressed in $E > ? \frac{GV}{m}$).
|
\boxed{$E > 30 \frac{GV}{m}$}
|
(1) Find the expression of the solar constant $S_0$.
(2) Calculate the value of $S_0$ (expressed in $W/m^2$).
|
\boxed{$S_0 = \sigma T_S^4 (\frac{R_S}{d})^2$}
|
(1) Find the expression of the Earth's temperature $T_{\mathrm{E}}$.
(2) Calculate the value of $T_{\mathrm{E}}$ (expressed in $\mathrm{K}$).
|
\boxed{$T_{\mathrm{E}} = (\frac{S_0}{4 \sigma})^{1/4} = \sqrt{\frac{R_S}{2d}} T_S$}
|
Find the function $f(x)$.
|
\boxed{$f(x) = 5(1-e^{-x})-x$}
|
(1) Calculate the numerical value of $x_{\mathrm{m}}$.
(2) From this value $x_{\mathrm{m}}$, find the value of $b$ (expressed in $\mathrm{nm} \cdot K$).
|
\boxed{$[2.89 \times 10^{6}, 2.90 \times 10^{6}]$}
|
(1) Find $\lambda_{\text{max}}^{\text{Sun}}$ for the Sun (expressed in $\mathrm{nm}$).
(2) Find $\lambda_{\text{max}}^{\text{Earth}}$ for the Earth (expressed in $\mathrm{nm}$).
|
\boxed{$[5.01 \times 10^{2}, 5.02 \times 10^{2}]$}
|
Use the Heisenberg's uncertainty principle to find $\Gamma$
|
\boxed{$\Gamma = \frac{1}{\tao}$}
|
In Bob's reference frame:
(1) Find the distance $l_B$ between two successive gifts sent by Alice.
(2) Find the time interval $\Delta t_{1}$ at which these gifts from Alice arrive at Bob's spaceship.
|
\boxed{$\Delta t_{1} = \frac{16}{35} \Delta t_0$}
|
At a given instant, Alice can see a number of gifts moving away from her and a number of gifts moving towards her. What is the ratio between these two numbers?
|
\boxed{18}
|
A symmetric spinning top,rotating clockwise at an angular frequency $\omega$, is placed upright in the center of a frictionless circular plate. The plate then begins to rotate counterclockwise at a constant angular velocity $\omega$. Assume the top's axis remains perfectly vertical and stable without any precession. From the perspective of an observer rotating with the plate, how does the top appear to rotate?
(A) The top appears stationary without any rotation.
(B) The top appears to rotate in the clockwise direction at an angular frequency $\omega$.
(C) The top appears to rotate in the clockwise direction at an angular frequency $2\omega$.
(D) The top appears to rotate in the counterclockwise direction at an angular frequency $\omega$.
(E) The top appears to rotate in the counterclockwise direction at an angular frequency ${2\omega}$.
|
\boxed{C}
|
If the lighter object is initially at rest, and the heavier object collides elastically with it, what is the approximate maximum fraction of the heavier object's kinetic energy that could be transferred to the lighter object?
(A) $m/M$.
(B) ${2m}/M$.
(C) ${4m}/M$.
(D) ${m}^2/{M}^2$.
(E) $2{m}^2/{M}^2$.
|
\boxed{C}
|
Now suppose that instead, the heavier object is initially at rest, and the lighter object collides elastically with it. What is the approximate maximum fraction of the lighter object's kinetic energy that could be transferred to the heavier object?
(A) $m/M$.
(B) ${2m}/M$.
(C) ${4m}/M$.
(D) ${m}^2/{M}^2$.
(E) $2{m}^2/{M}^2$.
|
\boxed{C}
|
A $50 g$ piece of clay is thrown horizontally with a velocity of $20 m/s$ striking the bob of a stationary pendulum with length $l = 1 m$ and a bob mass of $200 g$. Upon impact, the clay sticks to the pendulum weight and the pendulum starts to swing. What is the maximum change in angle of the pendulum?
(A) $\arccos(1/5)$.
(B) $\arcsin(7/10)$.
(C) $\arccos(2/3)$.
(D) $\arcsin(3/10)$.
(E) $\arctan(4/5)$.
|
\boxed{A}
|
There are three kinds of balls that can be launched in this set-up, all having the same radius $R$:
I. a regular tennis ball (a thin spherical shell of rubber) of mass $m_1$.
II. a solid wooden ball of mass $m_2$.
III. a solid rubber ball of mass $m_3$.
where $m_1 < m_2 < m_3$. All three types of ball emerge from the launcher with the same velocity. For which ball will the final velocity be highest?
(A) Ball I.
(B) Ball II.
(C) Ball III.
(D) Balls II and III.
(E) The final velocity will be the same for all three balls.
|
\boxed{D}
|
Two soap bubbles of radii $R_1 = 1 cm$ and $R_2 = 2 cm$ conjoin together in the air,such that a narrow bridge forms between them. Assuming the system starts in equilibrium, the bubbles are extremely thin, and that air can flow freely between the bubbles through the bridge, describe the evolution and final state of the bubbles.
(A) The smaller bubble will shrink and the larger bubble will grow.
(B) The larger bubble will shrink and the smaller bubble will grow.
(C) The bubbles will maintain their sizes.
(D) Air will oscillate between the two bubbles.
(E) Both bubbles will simultaneously shrink.
|
\boxed{A}
|
A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = -k\frac{x^2 + y^2}{2}$. The closest point to the origin $(x = 0, y = 0)$ during its motion was at a distance $d$, and the particle's speed at that point was $v \neq 0$. Which of the following statements is true regarding the path of the particle after a long time $t$ ($t \gg d/v$)?
(A) The particle's trajectory will be circular.
(B) The particle's trajectory will be asymptotic to a straight line pointing away from the origin.
(C) The particle will spiral outwards away from the origin.
(D) The particle will travel on a parabolic trajectory.
(E) The particle will spiral inwards towards the origin.
|
\boxed{B}
|
A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = {kxy}/2$. If the particle begins at the origin, then it is possible to displace it slightly in some direction, so that the particle subsequently oscillates periodically. What is the period of this motion?
(A) $2\pi\sqrt{m/{4k}}$.
(B) $2\pi\sqrt{m/{2k}}$.
(C) $2\pi\sqrt{m/k}$.
(D) $2\pi\sqrt{{2m}/k}$.
(E) $2\pi\sqrt{{4m}/k}$.
|
\boxed{D}
|
Near the ground, wind speed can be modeled as proportional to height above the ground. (This is a reasonable assumption for small heights.) A wind turbine converts a constant fraction of the available kinetic energy into electricity. The conditions are such that when operating at $10 m$ above the ground,the turbine delivers $15 kW$ of power. How much power would the same windmill deliver if it were operating at $20 m$ above the ground?
(A) $15 kW$.
(B) $21 kW$.
(C) $30 kW$.
(D) $60 kW$.
(E) $120 kW$.
|
\boxed{E}
|
The International Space Station orbits the Earth in a circular orbit $400 km$ above the surface,and a full revolution takes 93 minutes. An astronaut on a space walk neglects safety precautions and tosses away a spanner at a speed of $1 m/s$ directly towards the Earth. You may assume that the Earth is a sphere of uniform density. At which of the following five times will the spanner be closest to the astronaut?
(A) After 139.5 minutes.
(B) After 131.5 minutes.
(C) After 93 minutes.
(D) After 46.5 minutes.
(E) After 1 minute.
|
\boxed{C}
|
A student conducted an experiment to determine the spring constant of a spring using a ruler and two different weighing scales. The measured elongation of the spring was $1.5 cm$ ,and the smallest division on the ruler was $1 mm$ . The mass of the attached weight was measured using two different scales in the school laboratory,yielding values of $198 g$ and $210 g$ . The student also found that the local acceleration due to gravity in her city is given as $(9.806 \pm 0.001) m/s^2$. Calculate the percent error in measuring the spring constant.
(A) $2\%$.
(B) $4\%$.
(C) $8\%$.
(D) ${11}\%$.
(E) ${14}\%$.
|
\boxed{B}
|
It is possible to write the ring's magnetic moment $\vec{\mu}$ in terms of its angular momentum $\vec{L}$ as $\vec{\mu} = \gamma \vec{L}$. Find the constant $\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$.
|
\boxed{$\gamma = \frac{Q}{2M}$}
|
Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth.
|
\boxed{$P_0 = (1 - a) \pi R_E^2 F_s$}
|
Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. Express your answer in $K$.
|
\boxed{255}
|
Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\omega_{d}$?
|
\boxed{$\omega_d = \sqrt{k \frac{m_A + m_B}{m_A m_B}}$}
|
Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects.
|
\boxed{$E_p = \hbar \omega_d$}
|
What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \ll c$, where $c$ is the speed of light.
|
\boxed{$f - f_0 = \frac{v}{c} f_0$}
|
Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\infty$ to $\infty$.
|
\boxed{$C = \sqrt{\frac{m}{2 \pi k_B T}}$}
|
Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor.
|
\boxed{$p_2(f) = C \exp\left[-\frac{mc^2}{2k_B T} \left(\frac{f - f_0}{f_0}\right)^2\right]$}
|
Sketch $p_{2}(f)$ as a function of $f - f_{0}$, and determine the shift $f^{\star} - f_{0}$ at which $p_{2}(f^{\star})$ is a fraction $1 / e$ of its peak value, where $e$ is the natural number.
|
\boxed{$f^{\star} - f_0 = f_0 \sqrt{\frac{2k_B T}{mc^2}}$}
|
README.md exists but content is empty.
- Downloads last month
- 10