Rotational Kinetic Energy, Torque & Work (College Board AP® Physics 1: Algebra-Based): Exam Questions

A fairground wheel of radius $15\mathrm{m}$ and mass $4\times {10}^5\mathrm{kg}$ is mounted on the fixed, frictionless axle of a motor. The wheel can be modeled as a hoop of rotational inertia $M{R}^2$. The motor increases the angular speed of the wheel from rest to a maximum value of $0.35\mathrm{rad}/\mathrm{s}$ in $25.0\mathrm{seconds}$.

Which of the following is most nearly the average power output of the motor?

When a force of $60\mathrm{N}$ is applied normally to the end of the wrench to tighten a bolt, it turns through an angle of $45°$.

Which of the following is most nearly the work done on the bolt if the distance from the end of the wrench to the center of the bolt is $22\mathrm{cm}$?

The graph shows the net torque applied to a rotating body as a function of angular position.

Which of the following is most nearly the change in kinetic energy of the body after it has rotated through an angle of 35°?

Two wheels mounted on separate frictionless axles are joined by a loop of cord, as shown in the figure. The smaller wheel of radius $R$ has a rotational inertia of $I$, and the larger wheel of radius $2R$ has a rotational inertia of $16I$. The axis of the smaller wheel is attached to a motor that rotates it at angular speed $\omega$, which in turn causes the larger wheel to rotate.

Assuming the loop does not slip on the wheels, which of the following expressions represents the total kinetic energy of the system?

A cake of mass $8m$ and radius $R$ rotates on a decorating turntable of negligible mass at an angular velocity of $\omega$. The cake can be modeled as a solid disc of rotational inertia $\frac{1}{2}M{R}^2$ and rotational kinetic energy $K$. When four balls of frosting, each of mass $m$, are placed onto the outer edge of the cake, the new angular velocity of the system is $\frac{\omega }{2}$.

Which of the following correctly describes the new rotational kinetic energy of the system and the work done on the cake by the balls of frosting?

A uniform rod of length $L$ and mass $M$ is initially held vertically with one end resting on the horizontal ground, as shown in the figure. When the rod is released from rest, the upper end falls to the ground, and the lower end does not slip. The rotational inertia of the rod about one of its ends is $\frac{1}{3}M{L}^2$.

Which of the following expressions represents the linear velocity of the upper end of the rod immediately before it strikes the floor?

A mass $m$ is attached to a light cable which is wrapped around the rim of a cylindrical pulley of radius $R$, as shown in the figure. The cylinder rotates about an axle at its center with rotational inertia $\frac{1}{2}M{R}^2$. When the cylinder rotates, a constant frictional torque acts on the system. Mass $m$ is initially held at rest at a height $h$ above the ground. After the mass is released, it strikes the ground with speed $v$.

Which of the following expressions represents the work done by the system against friction as the mass $m$ falls through height $h$?

A uniform rod of mass $M$ and length $L$ is free to rotate about a frictionless pivot at its left end and held horizontally by a string of negligible mass at its right end, as shown in the figure. The string is cut and releases the rod, causing it to rotate about the axis and swing down from the horizontal position.

Which of the following graphs could represent the net torque $\tau$ on the rod as a function of the angular displacement $\theta$ from the initial horizontal position?

Two identical yo-yos are initially held at rest at the same height above the ground. The energy bar chart in the figure represents the rotational kinetic energy $K_r$ and translational kinetic energy $K_t$ of the yo-yos, and the gravitational potential energy $U_g$ of the yo-yo-Earth systems just before their release. The yo-yos are then released at the same instant, one is dropped without its string attached to anything, and the other one falls with its string tied to someone's hand, causing it to unwind and rotate.

Both yo-yos can be modeled as two solid discs, each of mass $M$, radius $R$, and rotational inertia $\frac{1}{2}M{R}^2$, which are separated by a thin string wound about an axle of radius $\frac{R}{2}$.

Which of the following graphs could represent the energy bar charts for the yo-yos just before they hit the ground?

A yo-yo can be modeled as two solid discs, each of mass $M$, radius $R$, and rotational inertia $\frac{1}{2}M{R}^2$, separated by a thin string wound about an axle of radius $r$. Initially, a yo-yo and its wound string are held at rest. When the yo-yo is released, it falls vertically and the string unwinds from the axle causing it to spin about its center of mass with linear speed $v$, as shown in the figure. The total length of the unwound taut string is $L$.

Which of the following expressions represents the linear speed of the center of mass of the yo-yo when the string is fully unwound?