A point on a disk rotates around the center axle of the disk. The table shows the angular speed of the point as a function of time. Which of the following graphs could represent the angular position of the point as a function of time?

A disk is fixed to a horizontal axle that extends between two supports, as shown in the figure. Frictional forces between the axle and the supports is not negligible. At time $$t_s$$  , the disk rotates about the center axle with an initial angular speed $$\omega_d$$  . A student measures the angular displacement $$\Delta\theta_o$$   of a point on the edge of the disk from time $$t_s$$   until the disk no longer rotates. The angular acceleration of the disk is determined to be $$\alpha_d$$  , and this value remains constant. Based on the data, if possible, how could the student predict the angular displacement of a point on the edge of the disk from time $$t_s$$   until the disk no longer rotates if the initial angular speed is increased to $$2\omega_d$$  ? Justify your selection.

A group of students must conduct an experiment to determine how the location of an applied force on a classroom door affects the rotational motion of the door. The rotational inertia of the door about its hinges is known. The initial angular velocity of the door is zero.

Which of the following lists what measuring devices the students need and the measurements they should take to collect the necessary data to test the relationship between a torque exerted on the door and the change in angular velocity of that object about the hinges of the door? Justify your selection.

An object of mass M hangs from a string that is looped around a pulley of negligible friction, as shown. The pulley has a mass 0.5M. The object is released from rest and it falls to the floor at time $$t_1$$  . Which of the following pairs of graphs best represents the angular speed as a function of time for the pulley and the vertical speed as a function of time for the falling object for a short time after it is released from rest?

A net torque is applied to the edge of a spinning object as it rotates about its internal axis. The table shows the net torque exerted on the object at different instants in time. How can a student use the data table to determine the change in angular momentum of the object from 0s to 6s? Justify your selection.

A uniform disk with mass M_o  and radius R is mounted on a vertical axis so that it can rotate freely in a horizontal plane. The rotational inertia of the disk is I_o. The disk is initially at rest. If force F_o is exerted tangentially to the rim of the disk for time interval Δt, the final angular momentum of the disk is L_o.

Consider a second disk that has the same mass M_o as the first disk, but its radius is 2R. The rotational inertia of this disk is 4I_o. If the same force F_o is exerted tangentially to the rim of the disk for the same time interval Δt, then the final angular momentum of the second disk would be

One end of a horizontal rod is connected to an axle that is connected to a motor that can be adjusted to change the angular speed of the rod as it rotates in a horizontal circle. Students must determine the change in angular momentum of the rod after 10s. The students use the following procedure.

1. Measure the length of the rod with a meterstick.

2. Ensure that the rod is at rest.

3. Start the stopwatch and simultaneously adjust the motor so that the rod rotates around the axle as the angular speed is slowly increased.

4. Record the angular speed of the rod when the stopwatch reads 10s.

The students are provided with the equation for the rotational inertia of the rod about one end, $$I=\frac{ML^2}{3}$$  . Which of the following steps should the students add to the procedure to ensure that the change in angular momentum of the rod can be determined?

Three disks are concentrically attached to one another, and four rods of negligible mass are attached to the outer disk. Identical objects of mass $$M_o$$   can be attached to the rods, and their positions on the rods can be adjusted. The disks, rods, and objects form a system that freely rotates around a central axis that is perpendicular to the plane of the page. The objects are initially a distance D away from the axis of rotation. A constant force $$F_o$$   is applied tangent to the second disk, as shown in the figure. How can the system be changed so that the change in angular momentum of the system per unit of time is increased?

A uniform rod is at rest on a horizontal surface. A student may launch a sphere of clay toward the rod along one of the three paths shown in the figure. Path X and path Z are directed toward the center of mass of the rod. In each case, the sphere of clay is launched with the same linear speed and sticks to the rod. In each case, the time of collision between the sphere of clay and the rod is time $$t_o$$  . A pivot is fixed to the end of the rod, representing the point at which the rod or clay-rod system may rotate. Frictional forces are considered to be negligible.

A sphere of clay travels toward the rod along path ZZ. A student must predict what will happen to the linear momentum and the angular momentum of the rod-sphere system as a result of the collision. Which of the following correctly predicts the change, if any, of these quantities?

A merry-go-round disk with a rotational inertia of I_d about its center spins around its center axle with an initial angular velocity of ω₀. A child is standing near the edge of the merry-go-round, as shown in Figure 1. The child’s rotational inertia about the center of the disk when near the edge is I₀. As the merry-go-round spins, the child moves closer to the center, as shown in Figure 2, until the disk rotates with an angular velocity of ω₁. Which of the following equations could a student use to determine the rotational inertia I_s of the child-merry-go-round system about the center axle immediately after the child has moved to the location shown in Figure 2? Justify your selection.

An isolated spherical star of radius R₀ rotates about an axis that passes through its center with an angular velocity of ω₀. Gravitational forces within the star cause the star’s radius to collapse and decrease to a value r₀ < R₀, but the mass of the star remains constant. A graph of the star’s angular velocity as a function of time as it collapses is shown. Which of the following predictions is correct about the angular momentum L of the star immediately after the collapse?