AP Physics C Mechanics 2.7 Friction

[Maximum number: 2]

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

In a different scenario, the centers of mass of the original disk and ring each have the same initial speed v as they did in the original scenario. The ramp is replaced by a new ramp on which the disk and the ring initially slip as they roll up the new ramp.

Indicate whether the magnitude of the kinetic frictional force exerted on the disk by the new ramp is greater than, less than, or equal to the magnitude of the kinetic frictional force exerted on the ring by the new ramp while both are slipping.

Briefly justify your answer.

Example Response
The forces of kinetic friction are equal. If slipping, the friction is kinetic, and the force of kinetic friction on the hoop and the disk are the same because the coefficient of kinetic friction is only dependent on the surfaces, and the normal forces on both are the same.

[Maximum number: 7]

Students design an experiment using blocks of adjustable mass to investigate friction using the setup shown. Block 1 of initial mass 0.44 kg is placed on a rough horizontal surface and connected by a string to block 2 of initial mass 0.20 kg. The string extends over a pulley that has negligible mass and friction.

(a)

Calculate the minimum value of the coefficient of static friction $\mu_{s}$ that would keep the two-block system at rest.

The coefficient of friction is such that when block 2 is released from rest, block 1 travels across the surface. The acceleration a of each block is recorded with motion detectors 1 and 2, as shown in the figure. The data for the motion detectors as functions of time t are shown on the graphs. For each motion detector, the positive direction is away from the detector.

Figure for Question (a) — AP Physics C: Mechanics

[ 2 ]

For correctly evaluating Newton's second law equations for the system at rest
$m_{2} g-f=\left(m_{1}+m_{2}\right) a=0 \therefore m_{2} g=f$

1 point

For correctly substituting for static friction into above equation:
$\begin{aligned} m_{2} g=f=\mu_{s} F_{N}=\mu_{s} m_{1} g \mu_{s}=\frac{m_{2}}{m_{1}}=\frac{(0.20 \mathrm{~kg})}{(0.44 \mathrm{~kg})}=0.45 \end{aligned}$

1 point

Total for part (a)

2 points

(b)

Calculate the coefficient of kinetic friction $\mu_{k}$ between block 1 and the table.

[ 3 ]

For correctly evaluating Newton's second law equation for block 1:
$T-f=m_{1} a$

1 point

For correctly evaluating Newton's second law equation for block 2:
$m_{2} g-T=m_{2} a$

Combining the two equations
$m_{2} g-f=\left(m_{1}+m_{2}\right) a \therefore f=m_{2} g-\left(m_{1}+m_{2}\right) a$

Scoring note: Both points are earned for a single correct Newton's second law equation for the two-block system.

1 point

For correctly substituting for kinetic friction into above equation
$\begin{aligned} f=\mu_{k} F_{N}=\mu_{k} m_{1} g=m_{2} g-\left(m_{1}+m_{2}\right) a \therefore \mu_{k}=\frac{m_{2} g-\left(m_{1}+m_{2}\right) a}{m_{1} g} \mu_{k}=\frac{(0.20 \mathrm{~kg})\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)-(0.44 \mathrm{~kg}+0.20 \mathrm{~kg})\left(2.3 \mathrm{~m} / \mathrm{s}^{2}\right)}{(0.44 \mathrm{~kg})\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)}=0.11 \end{aligned}$

1 point

Total for part (c)

3 points

(c)

Careful measurements determine that the coefficient of kinetic friction is larger than the value calculated in part (c). Does the following explanation sufficiently account for the observed discrepancy?
"The horizontal table was not perfectly level before the experiment was conducted. The observed difference in the angle accounts for the difference in the expected and calculated values of $\mu_{k}$ ." Yes No

Justify your answer.

The experiment is moved to a surface with negligible friction and run for eight trials. In each trial, the students vary the masses $m_{1}$ and $m_{2}$ of blocks 1 and 2, respectively, while keeping the total mass $\left(m_{1}+m_{2}\right)=0.64 \mathrm{~kg}$ constant. The data for the acceleration a of block 1 as a function of $m_{2}$ are shown on the graph below.

Figure for Question (c) — AP Physics C: Mechanics

[ 2 ]

AP Physics C: Mechanics 2.7 Kinetic and Static Friction Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)