A solid disk of mass \(m\), radius \(r\), and rotational inertia \(I = \frac{1}{2} mr^2\) is rolling across level ground without slipping. What is the ratio of the disk’s Rotational Kinetic Energy, \(KE_R\), to its translational Kinetic Energy, \(KE_T\)?

Use the graph below to answer Questions 2 and 3.

The torque exerted by a car engine on an axle is shown as a function of the angle, theta, that the axle has rotated through in the graph above. How much work is done by the engine on the axle between \(t = 4\) and \(t = 10\) seconds?

By how much does the Rotational Kinetic Energy of the axle change between \(t = 10\) seconds and \(t = 16\) seconds? Ignore any friction within the axle.

Use the graph below to answer Questions 4 and 5.

The torque exerted by a car engine on an axle is shown as a function of time in the graph above. Using the graph, which of the following can be determined without being given any other information for the time period shown?

If the rotational inertia of the axle is 50 kgm², and the axle is at rest at time \(t = 0\), what is the angular speed of the axle at \(t = 8\) seconds?

A uniform bar of length \(L\) is attached to a fixed axis located at its center of mass, which is located at point \(X\) in the diagram above. The bar has a rotational inertia of \(I_o\) around point \(X\). Determine the magnitude and direction of the angular acceleration, \(\alpha\), of the bar around the axis when forces are applied to the bar as shown in the diagram above.

A sphere of mass \(m\) rolls across a table with velocity \(v_o\) as shown above. Around which of the labeled points does the sphere have the greatest angular momentum?

Use the graph below to answer Questions 8 and 9.

External torques can be applied to a disk in order to change the angular momentum of the disk. The angular momentum of a disk for three time intervals is shown in the graph provided above. During \(t_1\), a single, external torque \(\tau_1\) is applied to the disk. During \(t_2\), the initial external torque \(\tau_1\) is applied, along with a frictional torque \(\tau_F\). During \(t_3\), a net torque \(\tau_3\) is applied to the disk.

How does the magnitude of \(\tau_F\) compare to the magnitude of \(\tau_1\)?

How does the magnitude of \(\tau_3\), the net torque exerted during \(t_3\), compare to the magnitude of \(\tau_1\), the net torque exerted during \(t_1\)?

A satellite orbiting the earth is rotating at a rate \(\boldsymbol{\omega}_o\) around an axis that passes through its center of mass. In this configuration, the satellite has rotational inertia \(I_o\). The satellite then unfolds two solar panels, increasing the rotational inertia to 1.4 \(I_o\) but causing no change in the location of the center of mass.

Which of the following describes both how the angular speed of the satellite changes, and why that change occurs?

I. The angular speed of the satellite increases so that angular momentum is conserved

II. The angular speed of the satellite remains constant so that angular momentum is conserved

III. The angular speed of the satellite decreases so that Rotational Kinetic Energy is conserved

IV. The angular speed of the satellite decreases so that angular momentum is conserved

Use the diagram below to answer Questions 11 and 12.

A person of mass \(M\) is standing at point \(A\) on a large disk that has rotational inertia \(I_o\) while the disk is spinning at an angular speed \(ꞷ_o\). At time \(t = 0\), the person walks at a constant linear speed along a radius toward point \(B\).

As the person is walking from \(A\) to \(B\), which of the following quantities remains constant?

A person of mass \(M\) is standing at point \(A\) on a large disk that has rotational inertia \(I_o\) while the disk is spinning at an angular speed \(ꞷ_o\). At time \(t = 0\), the person walks at a constant linear speed along a radius toward point \(B\).

As the person is walking from \(A\) to \(B\), which system does not have an external, unbalanced torque acting on it?

Use the diagram below to answer Questions 13–17.

Three different objects are placed at the top of identical ramps.

• In Case 1, a solid disk of mass M and radius R is placed at the top of a ramp where there is sufficient friction so that the disk rolls without slipping down the ramp.

• In Case 2, a solid disk of mass M and radius R is placed at the top of a ramp where the friction is insufficient so that the disk rolls and slips as it travels down the ramp.

• In Case 3, a block of mass M is placed at the top of the ramp, and there is no friction between the block and the ramp.

Which of the following correctly ranks the Cases 1, 2, and 3 from most to least translational Kinetic Energy gained as the objects travel from the top to the bottom of the ramp?

Three different objects are placed at the top of identical ramps.

• In Case 1, a solid disk of mass M and radius R is placed at the top of a ramp where there is sufficient friction so that the disk rolls without slipping down the ramp.

• In Case 2, a solid disk of mass M and radius R is placed at the top of a ramp where the friction is insufficient so that the disk rolls and slips as it travels down the ramp.

• In Case 3, a block of mass M is placed at the top of the ramp, and there is no friction between the block and the ramp.

Which of the following correctly ranks the three Cases from most to least total Kinetic Energy as the objects travel from the top to the bottom of the ramp?

Three different objects are placed at the top of identical ramps.

• In Case 1, a solid disk of mass M and radius R is placed at the top of a ramp where there is sufficient friction so that the disk rolls without slipping down the ramp.

• In Case 2, a solid disk of mass M and radius R is placed at the top of a ramp where the friction is insufficient so that the disk rolls and slips as it travels down the ramp.

• In Case 3, a block of mass M is placed at the top of the ramp, and there is no friction between the block and the ramp.

Which of the following correctly ranks the three Cases from most to least force of friction while the objects travel from the top to the bottom of the ramp?

Three different objects are placed at the top of identical ramps.

• In Case 1, a solid disk of mass M and radius R is placed at the top of a ramp where there is sufficient friction so that the disk rolls without slipping down the ramp.

• In Case 2, a solid disk of mass M and radius R is placed at the top of a ramp where the friction is insufficient so that the disk rolls and slips as it travels down the ramp.

• In Case 3, a block of mass M is placed at the top of the ramp, and there is no friction between the block and the ramp.

Which of the following correctly ranks the three Cases from most to least in terms of the magnitude of the Work done by friction on the object as the objects travel from the top to the bottom of the ramp?

Three different objects are placed at the top of identical ramps.

• In Case 1, a solid disk of mass M and radius R is placed at the top of a ramp where there is sufficient friction so that the disk rolls without slipping down the ramp.

• In Case 2, a solid disk of mass M and radius R is placed at the top of a ramp where the friction is insufficient so that the disk rolls and slips as it travels down the ramp.

• In Case 3, a block of mass M is placed at the top of the ramp, and there is no friction between the block and the ramp.

In which of the three Cases is there a linear relationship between \(v_{cm}\) and the rotational speed, \(\omega\), of the object as the objects travel down the ramp?

Use the Diagram below for Questions 18-21

The diagram above shows two different planetary orbital paths around the same star. Both orbits contain point A as shown. Which of the following quantities have the same magnitude in Orbit 1 and Orbit 2 as the planets pass point A?

Which of the following quantities is conserved in orbit 1, but is not conserved in orbit 2?

I. Kinetic Energy

II. Gravitation Potential Energy of the star-planet system

III. Mechanical Energy of the star-planet system

IV. Angular Momentum

The distance from the center of the star to the circular path in Orbit 1 is \(2R\), where \(R\) is the radius of the star. The mass of the star is \(M\), while the mass of the orbiting planet is \(m\). Calculate the total mechanical energy in this circular orbit in terms of \(M, m, R\), and \(G\).

What is the escape velocity for an object as it passes point \(A\) in Orbit 2?