Practice Test 8 - AP Physics C Mechanics

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. What is the rotational inertia of the system about an axis that passes through one of the masses and is perpendicular to the plane of the triangle?
2. A uniform rod of mass $M$ and length $L$ is pivoted at one end and oscillates as a physical pendulum. The rotational inertia of the rod about its end is $\frac{1}{3}ML^2$. What is the period of the rod for small-amplitude oscillations?
3. A gyroscope with its axis tilted precesses with a constant angular speed about a vertical axis. The gravitational force exerts a torque on the gyroscope, causing the precession. What is the work done by this gravitational torque during one complete precession cycle?
4. A uniform rod of length $L$ and mass $M$ is pivoted at its center. A force $F$ is applied perpendicularly to the rod at one end, and a force $2F$ is applied perpendicularly at the other end, in the opposite direction. The rotational inertia of the rod about its center is $I = \frac{1}{12}ML^2$. What is the magnitude of the initial angular acceleration of the rod?
5. A flywheel in the shape of a uniform disk has moment of inertia $I$ and is initially rotating with angular velocity $\omega_0$. A constant braking torque $\tau_b$ is applied until the flywheel stops. During the braking process, what is the average power dissipated?
6. Two satellites collided elastically in space: A ($200\,\text{kg}$) moved at $+5.0\,\text{m/s}$ and B ($200\,\text{kg}$) moved at $-2.0\,\text{m/s}$, with negligible external impulse. Determine the velocity of the second object after the collision.
7. Three particles are arranged in a straight line. Particle A has mass $2m$ at position $x = 0$, particle B has mass $m$ at position $x = L$, and particle C has mass $3m$ at position $x = 2L$. What is the distance from particle A to the center of mass of the system?
8. A 1000 kg drag racing car starts from rest while its engine delivers constant power of 80 kW for 5.0 s, neglecting losses. What kinetic energy does it gain?
9. Two skaters collide elastically: $m_1=50\,\text{kg}$ with $v_{1i}=+4.0\,\text{m/s}$ and $m_2=70\,\text{kg}$ with $v_{2i}=0$. Afterward $v_{2f}=+3.0\,\text{m/s}$. What is the final velocity of skater 1?
10. A particle moves in a conservative force field where the potential energy is given by $U(x) = ax^2 - bx^3$, where $a$ and $b$ are positive constants. The particle starts from rest at $x = 0$. What is the maximum value of $x$ the particle can reach if its total mechanical energy is $E$?
11. A solid ball starts from rest at height $H$ on an incline and rolls without slipping to the bottom. It then encounters a second, frictionless incline. What is the maximum vertical height $H'$ the ball reaches on the second incline?
12. Two disks on a common frictionless axle are brought into contact. Disk 1 has rotational inertia $I_1$ and is initially spinning. Disk 2 has rotational inertia $I_2$ and is initially at rest. They eventually rotate together at a common final angular velocity. Why is the total angular momentum of the two-disk system conserved during this process?
13. A force $F$ is required to stretch a single ideal spring with constant $k$ by a distance $x$. If two identical springs are connected in parallel, what total force is required to stretch the combination by the same distance $x$?
14. A particle's velocity changes from $\vec{v}_i = (3.0\hat{i} - 4.0\hat{j})$ m/s to $\vec{v}_f = (5.0\hat{i} + 4.0\hat{j})$ m/s. The average acceleration is in the same direction as the change in velocity, $\Delta\vec{v}$. What is the unit vector $\hat{u}$ in the direction of the average acceleration?
15. A wheel experiences net torque $\tau_{\text{net}}$; by $\tau=I\alpha$, how does increasing $\tau_{\text{net}}$ affect rotational motion if $I$ is constant?
16. A force F applied perpendicularly at a distance r from a pivot creates a torque of magnitude $\tau$. If the force is doubled to $2F$ and the distance from the pivot is halved to $r/2$, the magnitude of the new torque is:
17. A particle's position is described by the vector function $\vec{r}(t) = (t^3)\hat{i} + (2t^2)\hat{j}$, where $t$ is in seconds and position is in meters. What is the average velocity vector of the particle from $t = 0$ s to $t = 2$ s?
18. A book rests on a table, which stands on the floor. Consider the force that the book exerts on the table and the force that the table exerts on the book. According to Newton's third law, these forces:
19. A robotic arm rotates about a fixed pivot in a horizontal plane. A force of magnitude $F=40\,\text{N}$ is applied at a point $r=0.30\,\text{m}$ from the pivot. The force makes an angle $\phi=60^\circ$ with the arm (the radius vector from the pivot to the point of application). The applied force remains constant in magnitude and direction relative to the arm at that instant. Frictional effects are negligible, and only the torque from this force is considered. Use the standard torque magnitude relation $\tau=rF\sin\phi$. Using the given information, determine the torque produced about the pivot by the applied force.
20. The rotational inertia of a rigid body of mass $M$ about a given axis is often expressed as $I = Mk^2$, where $k$ is the radius of gyration. What is the physical significance of the radius of gyration $k$?

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