AP Physics C Mechanics 5.6: Newton’s Second Law in Rotation

Relate angular-velocity changes to net torque and rotational inertia using Newton’s second law in rotational form for a rigid system.

[Maximum number: 6]

A uniform disk and ring, each of mass M and radius R, roll without slipping along a horizontal

surface, as shown in Figure 1. The outer edges of the disk and ring are made of the same

material. The center of mass of the disk and the center of mass of the ring each initially move

with the same constant speed v.

The disk and the ring then smoothly transition to a ramp that is inclined at an angle $\theta$ above

the horizontal. Both the disk and the ring continue to roll without slipping as they move up the

ramp, as shown in Figure 2.

The ring travels a greater distance along the ramp than the disk travels before each

momentarily comes to rest.

Figure 1

Figure 2

(a)

While the disk and the ring are rolling on the ramp without slipping, the magnitudes of

the static frictional force exerted on the disk and on the ring by the ramp are $f_{\mathrm{D}}$ and $f_{\mathrm{R}}$ ,

respectively.

Indicate whether $f_{\mathrm{D}}$ is greater than, less than, or equal to $f_{\mathrm{R}}$ by writing one of the following.

- $f_{\mathrm{D}}>f_{\mathrm{R}}$

- $f_{\mathrm{D}}<f_{\mathrm{R}}$

- $f_{\mathrm{D}}=f_{\mathrm{R}}$

Justify your answer using qualitative reasoning beyond referencing equations.

[ 3 ]

A	For indicating $f_{\mathrm{D}}<f_{\mathrm{R}}$	Point A1
		For a justification that compares one of the following: - The motions of the disk and the ring using translational kinematics - The motions of the disk and the ring using rotational kinematics - The rotational inertias of the disk and the ring	Point A2
		For a justification that includes one of the following: - Reasoning that attempts Newton's second law in translational form - Reasoning that attempts Newton's second law in rotational form - Reasoning that attempts conservation of energy	Point A3
		Example Response The ring has a greater rotational inertia because it has more mass distributed towards the edge. So the ring travels farther and has less acceleration down the ramp. Because the ring and the disk have the same mass, the gravitational forces exerted on the shapes are the same. From Newton's second law, the ring must have less net force down the ramp and more friction up the ramp.

(b)

A cylinder has mass M, radius R, and rotational inertia I about its central axis. The cylinder

rolls without slipping up a ramp that is inclined at an angle $\theta$ above the horizontal.

Derive an expression for the magnitude of the static frictional force f exerted on the

cylinder by the ramp. Express your answer in terms of $M, R, I, \theta$ , and physical constants,

as appropriate. Begin your derivation by writing a fundamental physics principle or an

equation from the reference information.

[ 3 ]

\multirow[t]{3}{*}{B} & For a multistep derivation that includes Newton's second law in both translational and rotational forms & Point B1 \\ \hline &

For using the relationship $a=r \alpha$ in an attempt to solve a system of equations
Scoring Note: Responses that include conservation of energy may earn full credit.

& Point B3 \\ \hline & Example Response $\begin{aligned} & \alpha_{\mathrm{sys}}=\frac{\sum_{\tau}}{I_{\mathrm{sys}}} \\ & \left(\frac{a}{R}\right)=\frac{f R}{I} \\ & a=\frac{f R^{2}}{I} \end{aligned}$ $\begin{aligned} & \vec{a}_{\mathrm{sys}}=\frac{\sum_{\vec{F}}}{m_{\mathrm{sys}}} \\ & f-M g \sin \theta=-M a \end{aligned}$ $f-M g \sin \theta=-M \frac{f R^{2}}{I}$ $f+M \frac{f R^{2}}{I}=M g \sin \theta$ $f\left(1+\frac{M R^{2}}{I}\right)=M g \sin \theta$ $f=\frac{M g \sin \theta}{1+\frac{M R^{2}}{I}}$ & \\ \hline \end{tabular}

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Covered syllabus points

5.6 Newton’s Second Law in Rotational Form question 1

Question (a)

Question (b)