# A2L Item 211

Goal: Problem solving with rotational kinematics

Source: CT151.2S02-39

Two masses, attached to the ends of a rigid massless rod, are rotating
about pivot P as shown in the picture below. The mass two meters from P
has speed 0.5m/s. What is the acceleration of the mass one meter from
P? 1. 0.05 m/s2
2. 0.0625 m/s2
3. 0.125 m/s2
4. 0.250 m/s2
5. 0.5 m/s2
6. 1 m/s2
7. None of the above
8. Cannot be determined

### Commentary:

(2) Every one of the possible wrong responses indicates a common error
that students make. After the problem has been discussed it is useful to
have students find the acceleration of the mass at 2m and see that the
accelerations are in the same ratio as the velocities. Drawing vector
diagrams showing the Δv for each mass is useful for explaining this
relationship.

# A2L Item 189

Goal: Problem solving with dynamics

Source: UMPERG-ctqpe168 A
uniform disk with mass M and radius R rolls without slipping down an
incline 30° to the horizontal. The friction force acting through
the contact point is

1. 0
2. Mg/3
3. Mg/4
4. Mg/6
5. none of the above

### Commentary:

(4) This problem requires students to use the 2nd law written in
terms of the CM acceleration and the rotational dynamic relation written
about the CM or the contact point. In either case they also need the
geometric constraint for rolling. This is a difficult problem for
students requiring a lot of additional knowledge, such as the moment of
inertia for a disk and, depending upon solution method, the Parallel
Axis Theorem.

Having gone to the trouble of solving the problem it is best to make
sure that the students glean as much as they can. A good followup
question is which would have a larger friction force, a hoop, a disk or
a sphere. They may try to reason from the acceleration of these objects
that the larger the acceleration, the smaller the friction force. The
friction force depends upon the mass, however, and the question cannot
be answered without knowledge of the masses.

# A2L Item 175

Goal: Problem solving

Source: UMPERG-ctqpe135.1 A
disk, having radius R and mass M, is free to rotate about an axis
through its center. A massless string is wound around disk and attached
to mass m. The moment of inertia for a disk given by is
1/2(MR2). If M=4m what is the acceleration of mass m?

1. 0
2. g/2
3. g/8
4. g/5
5. g/3
6. None of the above
7. Cannot be determined

### Commentary:

(5) Students answering #2 are likely making the common mistake of
thinking that the tension in the string is mg.

# A2L Item 172

Goal: Hone the concept of torque

Source: UMPERG-ctqpe130 Given
F1 = 6N, and F2 = 8N, what is the total torque

1. 1.0 N-m, out
2. 0.7 N-m, in
3. 7.0 N-m, out
4. 1.0 N-m, in
5. 6.0 N-m, out
6. None of the above.

### Commentary:

(6) Many students use the origin rather than the point A. This provides
the opportunity to stress that torque is found with respect to a
specified point. Students using the right hand rule incorrectly may

# A2L Item 167

Goal: Problem solving

Source: UMPERG-ctqpe118 A mass
m slides down a frictionless track of radius R=0.5m. As the mass
reaches the bottom, relative to the center of curvature, its angular
velocity is most nearly:

6. Cannot be determined

### Commentary:

(1) The velocity near the bottom can be found using energy
conservation.

# A2L Item 108

Goal: Problem solving with rotational kinematics

Source: UMPERG-ctqpe110

A flywheel rotating about an axis through its center starts from rest,
rotates with constant angular acceleration for 2 seconds while making
one complete revolution and thereafter maintains constant angular
velocity. How long does it take the wheel to make a total of 6 full
revolutions?

1. 4 secs
2. 5 secs
3. 6 secs
4. 7 secs
5. 9 secs
6. None of above
7. Cannot be determined

### Commentary:

(4) This problem provides an excellent opportunity to discuss the power
of graphs for problem solving. Making a sketch of angular velocity vs.
time provides the easiest way to answer the problem. Conversely, an
algebraic solution is complicated.

# A2L Item 105

Goal: Problem solving

Source: UMPERG-ctqpe147 A hoop of mass 4 kg and radius r rolls
without slipping down an incline 30° to the horizontal. The hoop is
released from rest. What is the speed of the hoop after its center has
fallen a distance h?

1. (4g(h-r))1/2
2. (2gh)1/2
3. (gh)1/2
4. (0.5g(h+r))1/2
5. none of the above
6. cannot be determined

### Commentary:

(3.) Students should realize that the speed cannot depend upon the
radius. Answer #2 is the speed that a falling point mass would have and
the hoop must have less than that.

# A2L Item 104

Goal: Hone the concept of torque

Source: UMPERG-ctqpe131 1. There are no points having zero total torque.
2. There is one point having zero total torque.
3. There are many points having zero total torque.

### Commentary:

(3) Many students realize that there is at least one point, the
intersection of the lines of action of the two forces, having zero
torque. A drawing is usually sufficient to convince students that there
is an entire line along which the torque is zero.

# A2L Item 095

Goal: Problem Solving

Source: UMPERG-ctqpe164 A uniform disk with mass M and radius R sits at rest on an incline
30° to the horizontal. String is wound around disk and attached to
top of incline as shown. The string is parallel to incline. The
tension in the string is :

1. Mg
2. Mg/2
3. 2Mg/5
4. Mg/4
5. None of the above
6. Cannot be determined

### Commentary:

(4) This problem can be solved a variety of ways. The simplest method is
to balance torques about the contact point. This situation is an
points about which to write the rotational dynamics equation.

# A2L Item 094

Goal: Problem solving

Source: UMPERG-ctqpe162 A uniform disk with R=0.2m rolls without slipping on a horizontal
surface. The string is pulled in the horizontal direction with force
15N. The disk’s moment of inertia is 0.4 kg-m2. The friction
force on the disk is:

1. 0
2. 15N, to the right
3. 10N, to the left
4. 5N, to the right
5. 5N, to the left
6. None of the above
7. Cannot be determined