AP Physics C Mechanics 6.5: Rolling Without Slipping

Analyze rolling without slipping by connecting translational speed, angular speed, radius, contact conditions, and rotational energy.

[Maximum number: 3]

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.

(a)

Derive an expression for the time it takes the sphere to travel from Point A to Point B in terms of $v_{0}, \mu$ , and physical constants as appropriate.

[ 2 ]

(i) For indicating the linear speed is equal to $R \omega$ when slipping stops at Point B 1 point
Example Response

v=R \omega

For correctly substituting v and $\omega$ from parts (d)(i) and (d)(ii) 1 point
Scoring Note: Substituting the acceleration from part (d)(i) into a valid kinematic equation that includes time can earn this point.
Example Response

v_{0}-\mu g t=R \frac{5 \mu g}{2 R} t

Example Solution

\begin{aligned} & v=R \omega \\ & v_{0}-\mu g t=R \frac{5 \mu g}{2 R} t \\ & \therefore t=\frac{2 v_{0}}{7 \mu g} \end{aligned}

(b)

Derive an expression for the linear velocity of the sphere upon reaching Point B in terms of $v_{0}$ .

[ 1 ]

(ii) For correctly substituting the expression for time from (e)(i) into the expression for velocity in (d)(i)

1 point

Example Solution

\begin{aligned} & v=v_{0}-\mu g t \\ & v=v_{0}-\mu g \frac{2 v_{0}}{7 \mu g} \\ & \therefore v=\frac{5}{7} v_{0} \end{aligned}

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

Total for part (e)

for question 3

15 points

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6.5.B—Describe the motion of a system that is rolling without slipping question 1

Question (a)

Question (b)