AP Physics C Mechanics 6.4 Momentum Conservation Overview

Analyze angular momentum for a selected system and determine when external torque or angular impulse changes the total about an axis.

[Maximum number: 2]

A system consists of a small sphere of mass m and radius R at rest on a horizontal surface and a uniform rod of mass M=2 m and length $\ell$ attached at one end to a pivot with negligible friction, where $R \ll \ell$ . There is negligible friction between the surface and the sphere to the right of Point A and nonnegligible friction to the left of Point A. The rod is held horizontally as shown in Figure 1, then is released from rest. The total rotational inertia of the rod about the pivot is $\frac{1}{3} M \ell^{2}$ and the rotational inertia of the sphere about its center is $\frac{2}{5} m R^{2}$ . After the rod is released, the rod swings down and strikes the sphere head-on. As a result of this collision, the rod is stopped, and the ball initially slides without rotating to the left across the horizontal surface.

Derive an expression for the linear speed $v_{0}$ of the sphere immediately after colliding with the rod in terms of the length $\ell$ and physical constants as appropriate.

After sliding a short distance, at time t=0 the sphere encounters a region of the horizontal surface with a coefficient of kinetic friction $\mu$ , beginning at Point A as indicated in Figure 1. The sphere begins rotating while sliding and eventually begins rolling without sliding at Point B, also as indicated.

For using a correct expression for conservation of angular momentum 1 point
Example Response

\begin{aligned} & L_{\mathrm{i}}=L_{\mathrm{f}} \\ & I \omega=m v r \end{aligned}

For correctly substituting the expression for $\omega_{f}$ from part (a) 1 point
Example Response

\frac{2}{3} m \ell^{2} \sqrt{\frac{3 g}{\ell}}=m v_{0} \ell

Example Solution

\begin{aligned} & L_{\mathrm{i}}=L_{\mathrm{f}} \\ & I \omega=m v r \\ & \frac{2}{3} m \ell^{2} \sqrt{\frac{3 g}{\ell}}=m v_{0} \ell \\ & \therefore v_{0}=\sqrt{\frac{4}{3} g \ell} \end{aligned}

Scoring Note: The last equation is not needed for scoring the item but is presented for clarity.

[Maximum number: 1]

Figure for Question 6.4 Conservation of Angular Momentum question 2 — AP Physics C: Mechanics

3．A cylinder of mass M and radius R is released from rest at the top of an inclined plane，as shown above． The cylinder rolls without slipping down the incline．The rotational inertia of the cylinder is $M R^{2} / 2$ ．

（b）Is the angular momentum of the cylinder conserved as the cylinder rolls down the vertical distance h ？ Yes No Justify your answer．

All question bank results loaded

Covered syllabus points

6.4 Conservation of Angular Momentum question 1

6.4 Conservation of Angular Momentum question 2