AP Physics C Mechanics — Practice

All practice problems and solutions for AP Physics C Mechanics, organized by unit. Worked examples stay on the unit pages.

Auto-collected from the practice sections of each unit’s notes (scripts/build_practice.py). Edit the source notes, not this page.

Unit 1: 1D and 2D Kinematics

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Practice

Multiple Choice

1. A projectile is fired from height $h$ with speed $v_0$ at angle $\theta$ above horizontal. Air resistance is negligible. Which equation determines its time of flight if the ground is $y=0$?

(A) $0=h+v_0\sin\theta\,t-\dfrac{1}{2}gt^2$

(B) $0=v_0\cos\theta\,t-\dfrac{1}{2}gt^2$

(C) $h=v_0t-\dfrac{1}{2}gt^2$

(D) $0=v_0\sin\theta-gt$

2. A projectile is launched from level ground. At the top of its path, its speed is half its launch speed. What was the launch angle?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $75^\circ$

3. Two projectiles are launched from the same point with the same speed at complementary angles $\theta$ and $90^\circ-\theta$, where $0<\theta<45^\circ$. On level ground, the projectile launched at the larger angle has

(A) the same range and a longer flight time

(B) the same range and a shorter flight time

(C) a longer range and a longer flight time

(D) a shorter range and a shorter flight time

4. A particle has $x(t)=At^3-Bt$ with $A,B>0$. At the instant when the particle’s velocity is zero, its acceleration is

(A) zero

(B) $2\sqrt{3AB}$

(C) $6\sqrt{B/(3A)}$

(D) $6A\sqrt{B/(3A)}$

5. A particle moves in the plane with $x=bt$ and $y=ct^2-dt^3$. At the instant when $v_y=0$, the acceleration vector points

(A) purely horizontal

(B) upward

(C) downward

(D) tangent to the trajectory

6. A runner moves so that her speed depends on position according to $v=v_0+kx$, where $v_0,k>0$. Her acceleration as a function of position is

(A) $k$

(B) $k(v_0+kx)$

(C) $k/(v_0+kx)$

(D) $v_0+kx$

7. A particle has $v(t)=v_0-\beta t^2$ with $v_0,\beta>0$. Which expression gives the distance traveled from $t=0$ until the particle first stops?

(A) $\int_0^{\sqrt{v_0/\beta}}(v_0-\beta t^2)\,dt$

(B) $\int_0^{v_0/\beta}(v_0-\beta t^2)\,dt$

(C) $\int_0^{\sqrt{v_0/\beta}}\lvert -2\beta t\rvert\,dt$

(D) $v_0\sqrt{v_0/\beta}$

8. A particle moves along the $x$-axis with velocity $v(t)=v_0-\alpha t^2$, where $v_0,\alpha>0$. At what time is the particle’s displacement from its starting point greatest?

(A) $t=\sqrt{v_0/\alpha}$

(B) $t=v_0/\alpha$

(C) $t=\sqrt{v_0/(3\alpha)}$

(D) $t=2v_0/\alpha$

9. A boat always points directly across a river of width $W$ with speed $v_b$ relative to the water. The current is parallel to the banks and has speed $u(y)=u_0y/W$, where $y$ is distance across the river. Compared with a river whose current is everywhere $u_0/2$, the boat’s downstream drift is

(A) smaller

(B) the same

(C) larger

(D) impossible to compare without $v_b$

10. A projectile is launched from level ground and lands back at the same height a fixed horizontal distance $R$ away. The launch speed is increased while $R$ is kept the same. Compared with the original two possible launch angles, the new two possible launch angles

(A) move closer to $45^\circ$

(B) move farther from $45^\circ$

(C) both increase

(D) both decrease

11. A particle moves along the $x$-axis with acceleration $a(x)=\alpha x$ and starts at $x=x_0>0$ from rest. Which expression gives its speed at $x=2x_0$?

(A) $\sqrt{\alpha x_0^2}$

(B) $\sqrt{3\alpha x_0^2}$

(C) $\sqrt{4\alpha x_0^2}$

(D) $\sqrt{6\alpha x_0^2}$

12. A particle moves in one dimension with acceleration $a=-kv^2$ when $v>0$, where $k>0$. Which statement must be true while the particle is moving in the positive direction?

(A) The velocity-time graph is a straight line.

(B) The velocity decreases, but the magnitude of the slope decreases as the particle slows.

(C) The acceleration is constant and negative.

(D) Equal decreases in speed take equal amounts of time.

FRQ

1. A bead moves along a straight track with acceleration $a(x)=\alpha x-\beta$, where $\alpha$ and $\beta$ are positive constants. At $x=0$, the bead has speed $v_0$ in the positive direction.

   $(A)$ Derive an expression for $v^2$ as a function of $x$.

   $(B)$ Find the condition on $v_0$ for the bead to reach $x=\beta/\alpha$.

   $(C)$ If the bead turns around before reaching $x=\beta/\alpha$, determine the turning point.

   $(D)$ Explain how the result changes if the bead initially moves in the negative direction.

2. A projectile is launched from a cliff of height $H$ with initial speed $v_0$ at angle $\theta$ above horizontal. A horizontal wind causes constant acceleration $a_w$ in the same direction as the projectile’s horizontal velocity.

   $(A)$ Derive expressions for $x(t)$ and $y(t)$.

   $(B)$ Find an equation for the time when the projectile reaches the ground.

   $(C)$ Derive the horizontal distance from the base of the cliff where the projectile lands.

   $(D)$ Determine whether increasing $a_w$ changes the time of flight, and justify your answer.

3. A particle moves along the $x$-axis. From $t=0$ to $t=T$, its velocity is $v(t)=v_0(1-t/T)^2$. From $t=T$ to $t=2T$, its acceleration is constant and chosen so the particle returns to its starting position at $t=2T$.

   $(A)$ Find the displacement during the first interval.

   $(B)$ Determine the velocity at $t=T$.

   $(C)$ Find the constant acceleration during the second interval.

   $(D)$ Sketch the velocity-time graph, labeling intercepts and areas with signs.

Unit 2: Force and Translational Dynamics

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Practice

Multiple Choice

1. A block of mass $m$ rests on a small platform scale mounted on an incline of angle $\theta$. The wedge and scale are at rest, and static friction prevents slipping.

If the scale measures the normal force on the block, its reading is

(A) $mg\sin\theta$

(B) $mg\cos\theta$

(C) $mg\tan\theta$

(D) $mg$

2. A block is pressed against a vertical wall by a horizontal force $F$. The coefficient of static friction is $\mu_s$. The smallest $F$ that can keep the block from sliding is

(A) $mg$

(B) $\mu_s mg$

(C) $mg/\mu_s$

(D) $\mu_s/g$

3. Two blocks of masses $m$ and $2m$ are connected by a light string and pulled across a frictionless table by force $F$ applied to the $2m$ block. The tension in the string is

(A) $F/3$

(B) $F/2$

(C) $2F/3$

(D) $F$

4. A falling object experiences drag force $bv$ upward. Taking downward as positive, which differential equation describes the motion?

(A) $m\dfrac{dv}{dt}=mg+bv$

(B) $m\dfrac{dv}{dt}=mg-bv$

(C) $m\dfrac{dv}{dt}=bv-mg$

(D) $m\dfrac{dv}{dt}=-mg-bv$

5. A block of mass $m$ sits on a rough incline of angle $\theta$. A horizontal force $F$ pushes the block into the incline. Which change most directly increases the maximum possible static friction?

(A) Decreasing $F$

(B) Increasing $F$

(C) Decreasing $m$ while keeping $F$ fixed

(D) Making the incline frictionless

6. A pendulum bob hangs motionless relative to a train accelerating horizontally with magnitude $a$. If the string makes angle $\theta$ with the vertical and the tension is $T$, which pair of equations is consistent with the bob’s rest in the train’s frame?

(A) $T\sin\theta=ma$ and $T\cos\theta=mg$

(B) $T\cos\theta=ma$ and $T\sin\theta=mg$

(C) $T=mg$ and $\tan\theta=a/g$

(D) $T=ma$ and $\tan\theta=g/a$

7. A car travels over the top of a circular hill of radius $R$. At the top, the driver feels an apparent weight equal to one-third of their normal weight. The car’s speed is

(A) $\sqrt{gR/3}$

(B) $\sqrt{2gR/3}$

(C) $\sqrt{gR}$

(D) $\sqrt{4gR/3}$

8. A bead slides on a frictionless circular hoop in a vertical plane. At the side of the hoop, its speed is $v$. The normal force magnitude is

(A) $mg$

(B) $mv^2/R$

(C) $mg+mv^2/R$

(D) $\sqrt{(mg)^2+(mv^2/R)^2}$

9. An elevator accelerates upward with magnitude $a$. Inside it, a mass $m$ hangs from a spring scale while a horizontal force $F$ pulls the mass sideways so the supporting string makes angle $\phi$ with the vertical. The tension in the string is

(A) $m(g+a)$

(B) $\dfrac{m(g+a)}{\cos\phi}$

(C) $\dfrac{mg}{\cos\phi}$

(D) $m\sqrt{g^2+a^2}$

10. A block rests on a scale mounted on a wedge inclined at angle $\theta$. The wedge accelerates horizontally to the right with magnitude $a$, and the incline rises to the right. The block remains at rest relative to the scale. If the scale measures the normal force on the block, its reading is

(A) $m(g\cos\theta-a\sin\theta)$

(B) $m(g\cos\theta+a\sin\theta)$

(C) $m(g\sin\theta+a\cos\theta)$

(D) $m(g+a)\cos\theta$

11. A small mass moves in a vertical circle on a string of length $R$. Its speeds at the bottom and top are $v_b$ and $v_t$, and the corresponding string tensions are $T_b$ and $T_t$. Which relation follows from Newton’s second law in the radial direction?

(A) $T_b-T_t=\dfrac{m(v_b^2-v_t^2)}{R}+2mg$

(B) $T_b-T_t=\dfrac{m(v_b^2-v_t^2)}{R}$

(C) $T_b+T_t=\dfrac{m(v_b^2+v_t^2)}{R}$

(D) $T_b-T_t=2mg-\dfrac{m(v_b^2-v_t^2)}{R}$

12. A block of mass $m$ rests on the floor of an elevator that accelerates upward with magnitude $a_y$ while also accelerating horizontally with magnitude $a_x$. The block does not slip relative to the floor. The minimum coefficient of static friction required is

(A) $\dfrac{a_x}{g+a_y}$

(B) $\dfrac{a_x}{g}$

(C) $\dfrac{g+a_y}{a_x}$

(D) $\dfrac{\sqrt{a_x^2+(g+a_y)^2}}{g}$

FRQ

1. A block of mass $m$ is inside a box that accelerates horizontally with acceleration $a$. The block is pressed against the box’s vertical wall and does not slip. The coefficient of static friction between the block and wall is $\mu_s$.

   $(A)$ Draw a free-body diagram for the block in the ground frame.

   $(B)$ Derive the normal force exerted by the wall on the block.

   $(C)$ Determine the condition on $a$ for the block not to slide down.

   $(D)$ If the box also accelerates upward with acceleration $a_y$, derive the new no-slip condition.

2. A bead of mass $m$ slides without friction on a circular hoop of radius $R$ fixed in a vertical plane. At an angle $\theta$ measured from the bottom, the bead has speed $v$

   $(A)$ Draw a force diagram for the bead.

   $(B)$ Write Newton’s second law in the radial direction.

   $(C)$ Write Newton’s second law in the tangential direction.

   $(D)$ At angle $\theta$, determine the speed at which the bead would just lose contact with the hoop, if such a speed is possible.

3. A mass $m$ falls from rest through a fluid with drag force $F_d=bv$ upward. Take downward as positive.

   $(A)$ Write the differential equation for $v(t)$.

   $(B)$ Determine the terminal speed.

   $(C)$ Without solving fully for $v(t)$, determine whether the acceleration is increasing, decreasing, or constant as the object falls.

   $(D)$ Design a linear graph that could be used to determine $b$ from measurements of speed and acceleration.

Unit 3: Work, Energy, and Power

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Practice

Multiple Choice

1. A force is always perpendicular to a particle’s velocity. The force can change the particle’s

(A) speed but not direction

(B) direction but not speed

(C) kinetic energy only

(D) total mechanical energy only

2. A block slides up a rough incline and comes momentarily to rest. Compared with its mechanical energy at launch, its mechanical energy at the top is

(A) greater

(B) smaller

(C) the same

(D) zero

3. A force $F(x)=3x^2-2x$ acts on a particle from $x=0$ to $x=L$. The work done is

(A) $L^3-L^2$

(B) $3L^2-2L$

(C) $L^3+L^2$

(D) $3L^3-L^2$

4. If $U(x)=ax^4-bx^2$ with $a,b>0$, the force is

(A) $F_x=4ax^3-2bx$

(B) $F_x=-4ax^3+2bx$

(C) $F_x=ax^4-bx^2$

(D) $F_x=-a/x^4+b/x^2$

5. A particle moves in one dimension with potential energy $U(x)$. At a stable equilibrium,

(A) $U'=0$ and $U''>0$

(B) $U'=0$ and $U''<0$

(C) $U'>0$ and $U''=0$

(D) $U<0$ only

6. A block starts from rest at height $H$ above a horizontal spring, slides on a frictionless track, and compresses the spring a distance $x$. If the block instead starts from height $4H$, the new maximum compression is

(A) $x/2$

(B) $x$

(C) $2x$

(D) $4x$

7. A spring with constant $k$ is cut into two equal halves. One half is used as a spring. Compared with the original spring, the energy stored for the same stretch $x$ is

(A) half as large

(B) the same

(C) twice as large

(D) four times as large

8. A block moves through a region where a force $F(x)=F_0e^{-x/L}$ acts in the direction of motion. The work done from $x=0$ to $x=2L$ is

(A) $F_0L(1-e^{-2})$

(B) $2F_0L$

(C) $F_0L e^{-2}$

(D) $F_0/L$

9. A cart of mass $m$ moves under constant power $P$ from rest, with no resistive forces. Its speed after time $t$ is

(A) $Pt/m$

(B) $\sqrt{2Pt/m}$

(C) $2Pt/m$

(D) $\sqrt{Pt/(2m)}$

10. A satellite moves outward from radius $r$ to radius $2r$ around a planet of mass $M$. The work done by gravity during this motion is

(A) $-\dfrac{GMm}{2r}$

(B) $-\dfrac{GMm}{r}$

(C) $\dfrac{GMm}{2r}$

(D) zero, because gravity is perpendicular to orbital motion

11. A particle in potential $U(x)=\dfrac{A}{x^2}-\dfrac{B}{x}$, with $A,B>0$, has a stable equilibrium at

(A) $x=A/B$

(B) $x=2A/B$

(C) $x=B/A$

(D) $x=\sqrt{A/B}$

12. A projectile is launched upward from the surface of a planet of radius $R$ with speed $v_{\text{esc}}/2$. Neglect air resistance. Its maximum distance from the planet’s center is

(A) $4R/3$

(B) $3R/2$

(C) $2R$

(D) $4R$

FRQ

1. A block of mass $m$ starts from rest at height $H$ on a frictionless curved track, then crosses a rough horizontal patch of length $L$ with coefficient of kinetic friction $\mu_k$ before compressing a spring of constant $k$.

   $(A)$ Derive the speed of the block just before the rough patch.

   $(B)$ Determine the speed just after the rough patch.

   $(C)$ Find the maximum spring compression.

   $(D)$ Determine the condition on $H$ for the block to reach the spring.

2. A particle of mass $m$ moves in the potential $U(x)=ax^4-bx^2$, where $a,b>0$

   $(A)$ Find all equilibrium positions.

   $(B)$ Classify each equilibrium as stable or unstable.

   $(C)$ If the particle has total energy $E=0$, find its turning points.

   $(D)$ If the particle has total energy $E=0$, determine where its speed is greatest and justify your answer using the energy diagram.

3. A small spacecraft of mass $m$ moves radially away from a planet of mass $M$. Its engine supplies constant power $P$ for time $t_0$, starting from rest at radius $R$. Ignore air resistance and the changing mass of the spacecraft.

   $(A)$ Write an energy equation relating the spacecraft’s speed and radius after the burn.

   $(B)$ Determine the minimum engine energy needed for escape if the burn ends at radius $r_f$.

   $(C)$ Explain whether delivering the same energy quickly or slowly changes the escape condition in this idealized model.

   $(D)$ Identify one assumption in the model that would fail for a real rocket.

Unit 4: Linear Momentum and Impulse

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Practice

Multiple Choice

1. A net force on a particle varies as $F(t)=F_0(1-t/T)$ from $t=0$ to $t=T$. The impulse is

(A) $F_0T$

(B) $F_0T/2$

(C) $F_0/T$

(D) zero

2. A ball of mass $0.20\ \text{kg}$ hits a wall moving to the right at $15\ \text{m/s}$ and rebounds to the left at $10\ \text{m/s}$. If the contact time is $0.050\ \text{s}$, the magnitude of the average force exerted by the wall is

(A) $20\ \text{N}$

(B) $60\ \text{N}$

(C) $100\ \text{N}$

(D) $250\ \text{N}$

3. A system of particles has total mass $M$. Which equation remains true even if the particles collide inelastically with each other?

(A) $\sum\vec F_{\text{ext}}=M\vec a_{\text{cm}}$

(B) $\sum\vec F_{\text{int}}=M\vec a_{\text{cm}}$

(C) $K_i=K_f$

(D) $\vec r_{\text{cm}}=\vec 0$

4. A projectile explodes at the top of its path into two fragments of masses $m$ and $3m$. If the smaller fragment stops immediately after the explosion, the speed of the larger fragment immediately after is

(A) $v/3$

(B) $v$

(C) $4v/3$

(D) $3v$

5. Two skaters push off from rest on frictionless ice. One has three times the mass of the other. If no external horizontal force acts, the heavier skater’s kinetic energy is

(A) one-ninth the lighter skater’s kinetic energy

(B) one-third the lighter skater’s kinetic energy

(C) equal to the lighter skater’s kinetic energy

(D) three times the lighter skater’s kinetic energy

6. A force on a mass $m$ is $F(t)=F_0t/T$ from $t=0$ to $T$ and then $F(t)=F_0(2-t/T)$ from $t=T$ to $2T$. If the mass starts from rest, its speed at $t=2T$ is

(A) $F_0T/m$

(B) $F_0T/(2m)$

(C) $2F_0T/m$

(D) $\sqrt{F_0T/m}$

7. A stationary object explodes into three equal masses. Two pieces leave at speed $v$ with angle $120^\circ$ between their velocities. The third piece leaves with speed

(A) $0$

(B) $v$

(C) $\sqrt{3}v$

(D) $2v$

8. A mass $m$ moving right with speed $v$ collides elastically in one dimension with an initially stationary mass $3m$. After the collision, the velocity of the mass $m$ is

(A) $-v/2$

(B) $-v/3$

(C) $v/3$

(D) $v/2$

9. A mass $m$ with speed $5\ \text{m/s}$ elastically collides head-on with a mass $3m$ initially moving toward it at $1\ \text{m/s}$. The final velocity of the mass $m$ is

(A) $-4\ \text{m/s}$

(B) $-2\ \text{m/s}$

(C) $1\ \text{m/s}$

(D) $5\ \text{m/s}$

10. A cart moves to the right at $4\ \text{m/s}$ while sand leaks out vertically downward at rate $2\ \text{kg/s}$ relative to the ground. Ignoring external horizontal forces, the horizontal acceleration of the remaining cart-sand system is

(A) zero

(B) $2\ \text{m/s}^2$ to the right

(C) $2\ \text{m/s}^2$ to the left

(D) impossible to determine without the cart mass

11. A cart of initial mass $M$ and speed $v_0$ collects rain falling vertically at rate $\lambda$. Neglect horizontal external forces. Its speed after time $t$ is

(A) $v_0$

(B) $\dfrac{Mv_0}{M+\lambda t}$

(C) $v_0+\lambda t/M$

(D) $\dfrac{(M+\lambda t)v_0}{M}$

12. A rocket expels fuel backward at speed $u$ relative to the rocket. With no external force, the rocket’s speed change as its mass decreases from $M_i$ to $M_f$ is

(A) $u\ln(M_i/M_f)$

(B) $u\ln(M_f/M_i)$

(C) $u(M_i-M_f)$

(D) $u(M_f/M_i)$

FRQ

1. A cart of initial mass $M$ moves on a frictionless horizontal track with speed $v_0$. Sand falls vertically into the cart at constant rate $\lambda$.

   $(A)$ Derive the cart’s speed as a function of time.

   $(B)$ Determine the horizontal force the cart exerts on newly collected sand.

   $(C)$ Determine the rate at which mechanical energy is lost.

   $(D)$ Explain why horizontal momentum is conserved even though kinetic energy is not.

2. A block of mass $m$ moving with speed $v_0$ collides with and sticks to a block of mass $2m$ attached to a spring of constant $k$ on a frictionless track.

   $(A)$ Find the speed of the combined blocks just after the collision.

   $(B)$ Determine the maximum compression of the spring.

   $(C)$ Find the fraction of the initial kinetic energy lost in the collision.

   $(D)$ Describe how the answer changes if the collision is elastic instead.

3. A projectile of mass $3m$ moving horizontally at speed $v_0$ explodes into three fragments of equal mass. One fragment moves straight upward at speed $v_0$, and a second moves at angle $30^\circ$ below the original direction with speed $2v_0$.

   $(A)$ Determine the velocity components of the third fragment.

   $(B)$ Determine the speed of the third fragment.

   $(C)$ Compare the total kinetic energy before and after the explosion.

   $(D)$ Explain what supplied the change in kinetic energy.

Unit 5: Torque and Rotational Dynamics

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Practice

Multiple Choice

1. The perpendicular-axis theorem applies to

(A) any three-dimensional rigid body

(B) point masses only

(C) flat laminae

(D) rolling objects only

2. A point mass $m$ is attached to the end of a massless rod of length $L$. About an axis perpendicular to the rod through a point $L/3$ from the mass, its moment of inertia is

(A) $mL^2$

(B) $mL^2/9$

(C) $4mL^2/9$

(D) $mL^2/3$

3. Two forces of magnitude $F$ are applied to the end of a rod of length $L$ pivoted at the other end. One force is perpendicular to the rod, and the other makes angle $\theta$ with the rod in the opposite rotational sense. The net torque magnitude about the pivot is

(A) $FL(1-\sin\theta)$

(B) $FL(1-\cos\theta)$

(C) $FL\sin\theta$

(D) $FL\cos\theta$

4. A disk and a hoop have the same mass and radius. The same torque is applied to each from rest for the same time. The disk’s final angular speed is

(A) larger than the hoop’s

(B) smaller than the hoop’s

(C) equal to the hoop’s

(D) impossible to compare without the torque value

5. A massive pulley of radius $R$ and rotational inertia $I$ has two tensions $T_1$ and $T_2$ applied by a non-slipping string. Its angular acceleration is

(A) $\dfrac{(T_2-T_1)R}{I}$

(B) $\dfrac{T_1+T_2}{IR}$

(C) $\dfrac{I}{(T_2-T_1)R}$

(D) $\dfrac{(T_2-T_1)}{IR}$

6. A uniform disk of mass $M$ and radius $R$ rotates about an axis perpendicular to its face and passing through a point halfway between its center and rim. Its moment of inertia is

(A) $\dfrac{1}{2}MR^2$

(B) $\dfrac{3}{4}MR^2$

(C) $MR^2$

(D) $\dfrac{3}{2}MR^2$

7. A uniform rod of length $L$ is pivoted at one end and held horizontally by a vertical string attached to the other end. A mass $m$ hangs from the rod at distance $2L/3$ from the pivot. The rod has mass $M$. The string tension is

(A) $Mg+mg$

(B) $Mg/2+2mg/3$

(C) $Mg+mg-T$ for some tension $T$

(D) zero

8. A rigid body is in static equilibrium under exactly three nonparallel forces. Which statement must be true?

(A) The forces are parallel.

(B) The lines of action pass through a common point.

(C) The forces have equal magnitudes.

(D) The net torque is nonzero.

9. A ladder leans against a frictionless wall and rests on a rough floor. A person climbs upward along the ladder. Before slipping occurs, the horizontal force from the wall

(A) decreases

(B) increases

(C) stays constant

(D) is always zero

10. A yo-yo unwinds from rest without slipping. If its axle radius is $r$ and rotational inertia is $I$, the tension is less than $mg$ because

(A) the string stretches

(B) gravity must both translate and rotate the yo-yo

(C) the net force on the yo-yo is zero

(D) mechanical energy is not conserved

11. A thin rod of length $L$ has linear density $\lambda(x)=Cx$ measured from one end. Its moment of inertia about that end is

(A) $\dfrac{1}{2}ML^2$

(B) $\dfrac{2}{3}ML^2$

(C) $\dfrac{1}{3}ML^2$

(D) $\dfrac{1}{4}ML^2$

12. A horizontal rod of length $L$ is hinged to a wall and held by a cord making angle $\theta$ with the rod. Masses $m$ and $2m$ hang from the rod at distances $L/4$ and $3L/4$ from the hinge. Neglect the rod’s mass.

The tension in the cord is

(A) $\dfrac{7mg}{4\sin\theta}$

(B) $\dfrac{7mg}{4\cos\theta}$

(C) $\dfrac{5mg}{4\sin\theta}$

(D) $\dfrac{3mg}{2\sin\theta}$

FRQ

1. A nonuniform rod of length $L$ and mass $M$ has density $\lambda(x)=Cx^2$ measured from the left end. It is pivoted at the left end and held horizontally by a vertical string at the right end.

   $(A)$ Determine $C$ in terms of $M$ and $L$.

   $(B)$ Find the rod’s center of mass.

   $(C)$ Determine the tension in the string.

   $(D)$ Determine the horizontal and vertical hinge force components.

2. Two blocks of masses $m_1$ and $m_2$ are connected by a light string over a pulley modeled as a disk of mass $M$ and radius $R$. The string does not slip and $m_2>m_1$.

   $(A)$ Draw force diagrams for the blocks and a torque diagram for the pulley.

   $(B)$ Derive the acceleration of the blocks.

   $(C)$ Find both string tensions.

   $(D)$ Determine the limiting acceleration as $M\to 0$ and explain why it makes sense.

3. A rigid bar is pivoted at one end and released from rest at angle $\theta_0$ above the horizontal. A small mass $m$ is attached at the free end, and the bar itself has mass $M$ and length $L$.

   $(A)$ Write the moment of inertia of the system about the pivot.

   $(B)$ Determine the net torque about the pivot at the instant the system is released.

   $(C)$ Determine the initial angular acceleration.

   $(D)$ Determine the initial tangential acceleration of the attached mass and state its direction.

Unit 6: Energy and Momentum of Rotating Systems

Full notes →

Practice

Multiple Choice

1. Static friction does no work on a rigid object rolling without slipping on a fixed surface because

(A) the contact point is instantaneously at rest

(B) friction is always zero

(C) the center of mass is at rest

(D) rotational kinetic energy is constant

2. A torque $\tau(t)=\tau_0t/T$ acts on a disk from $t=0$ to $t=T$. The angular impulse is

(A) $\tau_0T$

(B) $\tau_0T/2$

(C) $\tau_0/T$

(D) $I\tau_0T$

3. A central force always points along $\vec r$. Therefore, for motion under a central force,

(A) angular momentum about the force center is conserved

(B) mechanical energy is always conserved

(C) speed is always constant

(D) the orbit must be circular

4. A rolling hoop and rolling disk have the same mass, radius, and center-of-mass speed. The hoop has

(A) more total kinetic energy

(B) less total kinetic energy

(C) the same total kinetic energy

(D) no rotational kinetic energy

5. A hoop, disk, and solid sphere with the same mass and radius roll without slipping down the same incline. The object with the largest acceleration is the

(A) hoop

(B) disk

(C) solid sphere

(D) all tie

6. A solid sphere rolls without slipping down an incline from height $H$. A block slides frictionlessly from the same height. The ratio of the sphere’s translational speed at the bottom to the block’s speed at the bottom is

(A) $\sqrt{5/7}$

(B) $\sqrt{2/5}$

(C) $\sqrt{7/5}$

(D) $1$

7. A rigid object rolls without slipping with center-of-mass speed $v$. Its total kinetic energy is $K=\dfrac{3}{4}Mv^2$. If its radius is $R$, its moment of inertia about its center is

(A) $\dfrac{1}{4}MR^2$

(B) $\dfrac{1}{2}MR^2$

(C) $MR^2$

(D) $\dfrac{3}{2}MR^2$

8. A wheel rolls without slipping up a rough incline. Static friction is present but there is no slipping or other dissipation. Which quantity is conserved during the upward motion?

(A) translational kinetic energy only

(B) rotational kinetic energy only

(C) total mechanical energy

(D) angular momentum about the center only

9. A disk spins freely on a frictionless axle. A student drops clay onto the disk at radius $R/2$, where it sticks. During the collision,

(A) angular momentum about the axle is conserved but rotational kinetic energy decreases

(B) rotational kinetic energy is conserved but angular momentum decreases

(C) both angular momentum and rotational kinetic energy are conserved

(D) neither angular momentum nor rotational kinetic energy is conserved

10. A person sits on a spinning stool holding two masses. Pulling the masses inward increases angular speed because

(A) angular momentum is conserved while moment of inertia decreases

(B) kinetic energy is conserved while moment of inertia decreases

(C) torque from gravity increases

(D) the masses lose angular momentum to the stool

11. A satellite in an elliptical orbit is closest to the planet at periapsis. From periapsis to apoapsis, its angular momentum about the planet

(A) increases

(B) decreases

(C) remains constant

(D) becomes zero at apoapsis

12. A planet of mass $m$ moves in a circular orbit of radius $r$ around a star of mass $M$. If the star’s mass were replaced by $4M$ while $r$ stayed the same, the planet’s angular momentum magnitude would be multiplied by

(A) $1/2$

(B) $1$

(C) $2$

(D) $4$

FRQ

1. A solid sphere rolls without slipping down an incline of angle $\theta$ from rest.

   $(A)$ Derive its center-of-mass acceleration.

   $(B)$ Determine the static friction force and its direction.

   $(C)$ Find the translational and rotational kinetic energies after descending height $h$.

   $(D)$ Compare the result with a hoop released from the same height.

2. A disk of rotational inertia $I_0$ spins freely with angular speed $\omega_0$. A small block of mass $m$ initially at the center slides outward along a frictionless radial slot and latches at radius $R$.

   $(A)$ Determine the final angular speed.

   $(B)$ Determine the change in rotational kinetic energy.

   $(C)$ Explain where the missing mechanical energy goes during the latch.

   $(D)$ If the block is pulled inward slowly by an internal mechanism instead, explain whether work must be done.

3. A satellite of mass $m$ moves in an elliptical orbit around a planet of mass $M$. Its periapsis and apoapsis distances are $r_p$ and $r_a$.

   $(A)$ Use angular momentum conservation to relate $v_p$ and $v_a$.

   $(B)$ Use mechanical energy conservation to solve for $v_p$.

   $(C)$ Determine $v_a$.

   $(D)$ Explain why the satellite moves fastest at periapsis.

Unit 7: Oscillations

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Practice

Multiple Choice

1. A mass $m$ hangs from a spring of constant $k$ in a uniform gravitational field. It oscillates vertically about equilibrium with amplitude $A$. Which quantity depends on $g$?

(A) The angular frequency

(B) The period

(C) The equilibrium extension

(D) The speed at the equilibrium point measured relative to the oscillation amplitude

2. A pendulum clock is taken to a planet where the gravitational field strength is $g/4$. To keep the same small-angle period, the pendulum length should be changed from $L$ to

(A) $4L$

(B) $2L$

(C) $L/2$

(D) $L/4$

3. A mass on a spring is released from rest at $x=A$. When it first reaches $x=A/3$, what fraction of the total mechanical energy is kinetic?

(A) $1/9$

(B) $2/3$

(C) $8/9$

(D) $\sqrt{8}/3$

4. A block of mass $m$ is attached to a spring of constant $k$ on a frictionless horizontal surface. A small constant horizontal force $F_0$ is then applied and left on. Compared with the original oscillator, the new motion has

(A) the same angular frequency and an equilibrium shifted by $F_0/k$

(B) angular frequency $\sqrt{(k+F_0)/m}$ and the same equilibrium

(C) angular frequency $\sqrt{k/(m+F_0/g)}$ and an equilibrium shifted by $F_0/k$

(D) no simple harmonic motion because the net force is not proportional to $x$

5. A bead slides without friction on a circular hoop of radius $R$ in a vertical plane. Near the bottom of the hoop, the coordinate along the arc is $s=R\theta$. The bead’s small-oscillation angular frequency is

(A) $\sqrt{g/R}$

(B) $\sqrt{R/g}$

(C) $g/R$

(D) $\sqrt{2g/R}$

6. A mass $m$ is attached between two horizontal springs with constants $k_1$ and $k_2$ on a frictionless track, one spring on each side. Both springs are relaxed when the mass is at $x=0$. If the mass is displaced slightly, its angular frequency is

(A) $\sqrt{\dfrac{k_1+k_2}{m}}$

(B) $\sqrt{\dfrac{k_1k_2}{m(k_1+k_2)}}$

(C) $\sqrt{\dfrac{k_1-k_2}{m}}$

(D) $\sqrt{\dfrac{k_1}{m}}+\sqrt{\dfrac{k_2}{m}}$

7. A block attached to a spring oscillates on a frictionless table. The block is replaced by two identical blocks glued together, and the amplitude is doubled. The maximum acceleration changes by a factor of

(A) $1/\sqrt{2}$

(B) $1/2$

(C) $\sqrt{2}$

(D) $1$

8. A mass $m$ on a vertical spring oscillates about its equilibrium position with period $T$. At the instant the mass passes through equilibrium moving downward, a second identical mass is gently attached. Immediately after attachment, the new equilibrium position is

(A) unchanged

(B) lower by $mg/k$

(C) lower by $2mg/k$

(D) higher by $mg/k$

9. A pendulum bob of mass $m$ and length $L$ is also attached to a horizontal spring of constant $k$ that is relaxed when the bob hangs vertically. For small angles, compared with the same pendulum without the spring, the period is

(A) larger

(B) smaller

(C) unchanged

(D) zero because the forces cancel

10. A particle moves in the potential $U(x)=\dfrac{1}{2}kx^2+\epsilon x^4$, where $k,\epsilon>0$. For sufficiently small oscillations about $x=0$, the angular frequency is

(A) $\sqrt{k/m}$

(B) $\sqrt{(k+4\epsilon)/m}$

(C) $\sqrt{\epsilon/m}$

(D) dependent on amplitude even in the small-amplitude limit

11. A particle moves near $x=0$ in the potential $U(x)=U_0+ax^2+bx^3+cx^4$, where $a>0$. For sufficiently small oscillations, the angular frequency is

(A) $\sqrt{a/m}$

(B) $\sqrt{2a/m}$

(C) $\sqrt{6b/m}$

(D) $\sqrt{12c/m}$

12. A solid cylinder of mass $M$ and radius $R$ is attached at its center to a horizontal spring of constant $k$ and rolls without slipping. Its angular frequency is

(A) $\sqrt{k/M}$

(B) $\sqrt{2k/M}$

(C) $\sqrt{2k/(3M)}$

(D) $\sqrt{3k/(2M)}$

FRQ

1. A solid cylinder of mass $M$ and radius $R$ rests on a rough horizontal surface and rolls without slipping. A light spring of constant $k$ is attached to the cylinder’s center, and the other end is fixed to a wall. The cylinder is displaced a small distance $A$ from equilibrium and released from rest.

   $(A)$ Using energy, derive an expression for the angular frequency of the oscillation in terms of $M$ and $k$.

   $(B)$ Determine the maximum static friction force needed during the motion.

   $(C)$ Find the minimum coefficient of static friction required for rolling without slipping for the entire motion.

   $(D)$ Suppose the cylinder is replaced by a thin hoop with the same $M$ and $R$. Without redoing the full calculation, determine whether the period increases, decreases, or stays the same, and justify your answer.

2. A bead of mass $m$ slides without friction on a rigid circular wire of radius $R$ fixed in a vertical plane. The bead is also attached to a light spring of constant $k$ and negligible relaxed length whose other end is fixed at the top of the circle. Let $\theta$ be the bead’s angular displacement from the bottom of the circle.

   $(A)$ Write the bead’s gravitational potential energy and spring potential energy as functions of $\theta$, taking the bottom of the circle as zero gravitational potential.

   $(B)$ Find the condition on $k$ and $R$ for the bottom of the circle to be a stable equilibrium.

   $(C)$ For small oscillations about the bottom, derive the angular frequency in terms of $m$, $g$, $R$, and $k$.

   $(D)$ Describe qualitatively how the equilibrium position changes if the spring constant is made very large.

3. A student studies a cart-spring oscillator on a horizontal track. The cart of mass $M$ has a small block of mass $m$ resting on top of it. The coefficient of static friction between the block and cart is $\mu_s$. The cart is pulled to amplitude $A$ and released from rest; the block does not slip at first.

   $(A)$ Derive the period of the combined motion while the block does not slip.

   $(B)$ Determine the maximum amplitude $A_{\max}$ for which the block can remain at rest relative to the cart throughout the motion.

   $(C)$ The student measures the period for several added top-block masses $m$. Describe a graph that could be used to determine the spring constant $k$ from the data, including what should be plotted on each axis.

   $(D)$ If the block begins to slip near the endpoints of the motion, explain whether the measured period should be expected to match the expression from part $(A)$. Your explanation should refer to the forces on the two objects, not just energy loss.