# A2L Item 087

Goal: Hone angular kinematic quantities and distinguish them from linear kinematic quantities.

Source: UMPERG

A mass moves in a circle with uniformly increasing anglular velocity.

As the angular velocity ω increases, the linear acceleration of
the mass has…

1. constant magnitude and direction.
2. constant magnitude, changing direction.
3. changing magnitude, constant direction.
4. both magnitude and direction changing.

### Commentary:

(4) This requires exploration. Some students may think that the
direction is changing because the acceleration points toward the center
of the circle. They may be unaware that there is also a component of the
acceleration in the tangential direction.

and that ‘towards the center’ is a direction.

# A2L Item 086

Goal: Hone angular kinematic quantities and distinguish them from linear kinematic quantities.

Source: UMPERG

A mass moves in a circle with uniformly increasing angle.

As the angle θ increases, the linear acceleration of the mass has

1. constant magnitude and direction.
2. constant magnitude, changing direction.
3. changing magnitude, constant direction.
4. both magnitude and direction changing.

### Commentary:

(2) Students have a lot of difficulty reconciling linear kinematics with
angular kinematics. Unless shown how to take derivatives in polar
coordinates, or shown how to represent rotational kinematic quantities
as vectors, students can only memorize specific relationships.

Some students may answer (1) thinking that ‘towards the center’ is a
direction.

# A2L Item 083

Goal: Reasoning with dynamics.

Source: UMPERG

A child stands on a spinning disk. Suppose that there is friction
between the child’s shoes and the surface of the disk. While holding a
rock the child stands at the largest radius possible for the current
angular velocity without slipping. After releasing the rock, the child
will…

1. lose traction and slide off the disk.
2. lose traction, slide towards center.
3. remain on disk and able to move out.
4. remain on disk, unable to move out.

### Commentary:

(4) There should be no consequence of dropping the rock. Because the
normal force changes, so does the friction force. The new friction force
is still able to provide the necessary centripetal force for the
circular motion.

# A2L Item 082

Goal: Problem solving

Source: UMPERG

A bug sits on a disk at a point 0.5 m from the center. If the
coefficient of friction between the bug and disk is 0.8, the maximum
angular velocity the disk can have before the bug slips off the disk is
most nearly:

5. None of the above
6. Cannot be determined

### Commentary:

(2) Some students may respond (6) thinking that the mass of the bug is
needed for solution.

# A2L Item 080

Goal: Problem solving using conservation of angular momentum.

Source: UMPERG

A bug walks on a freely rotating disk. Given: Mbug=0.05 kg,
Idisk=0.03 kg-m2, Rdisk=0.5 m, and
ωo= 2 rad/s when bug is 0.1 m from center. The bug
crawls to 0.4 m from the disk’s center. The final ω is most
nearly:

6. None of the above
7. Cannot be determined

### Commentary:

(1) This is a very traditional conservation of angular momentum problem.
The only difficulty is due to the presence of extraneous information.
Some students may use conservation of energy.

# A2L Item 078

Goal: Hone the concepts of speed and velocity.

Source: UMPERG

The radius of the Earth is 6,400 km. The speed and direction would you
have to travel along the equator to make the sun appear fixed in the sky
is most nearly

1. 1680 km/hr, East
2. 840 km/hr, East
3. 533 km/hr, East
4. 267 km/hr, East
5. 267 km/hr, West
6. 533 km/hr, West
7. 840 km/hr, West
8. 1680 km/hr, West
9. Cannot be determined

### Commentary:

(8) You would attempt to remain underneath the sun as it traveled from
East to West. Some students may be confused by the tilt of the Earth’s
axis and think that the Sun could not remain fixed in the sky if you
were constrained to move along the equator. These students would likely

#### Background

Students should be able to determine the speed and direction even if
they do not yet have a solid grasp of velocity as a vector.

#### Questions to Reveal Student Reasoning

What is the circumference of the Earth? Does everyone on the Earth
travel at the same speed?

#### Suggestions

Build a simple model. Most students can readily grasp the result when
the Earth’s axis is perpendicular to the plane of the Earth’s orbit. A
model helps them understand that the tilt of the axis doesn’t matter.

# A2L Item 069

Goal: Problem solving.

Source: UMPERG-ctqpe146

A
child having mass 32kg is standing at the rim of a rotating disk of
radius 1.5m. The disk is free to rotate without friction. The disk has
moment of inertia I = 125kg-m2 and is initially at rest. The
child throws a rock of mass 4kg in the forward tangential direction as
shown in the figure with a speed of 5m/s. The final angular speed of
the disk is most nearly

8. None of the above
9. Cannot be determined

### Commentary:

(7) is the most appropriate response. Initially there is no angular
momentum in the system. The child and disk must rotate clockwise to
balance the angular momentum of the rock. Some students may forget to
add the moment of inertia of child to that of the disk.

#### Background

Throwing the rock tangentially gives the rock angular momentum relative
to the fixed center of the disk. The disk-child system must have an
angular momentum which is the negative of that of the rock. Thus, it is
possible to find the angular velocity. The angular velocity is
negative, i,e, into the page.

#### Questions to Reveal Student Reasoning

Does the total system have angular momentum just before the rock is
thrown? just after it is thrown?

Does the rock have angular momentum just before it is thrown? just
after it is thrown?

What happens to the child upon throwing the rock? Does the child move?
How?

#### Suggestions

Have students relate their answer to this question to items 67 and 68.

# A2L Item 068

Goal: Problem solving and developing strategic knowledge.

Source: UMPERG-ctqpe145

You are given this problem:

A
child is standing at the rim of a rotating disk holding a rock. The
disk rotates without friction. The rock is thrown in the RADIAL
direction [path (5)] at the instant shown. You are given:

• Mass of the child
• Mass of the thrown rock
• Velocity of the rock
• Initial angular speed of the system

You want to find the final angular speed of the disk and child.

What principle would you use to solve the problem MOST EFFICIENTLY?

1. Kinematics only
2. F= ma or Newton’s laws
3. Work-Kinetic Energy theorem
4. Impulse-Momentum theorem
5. Angular Impulse-Angular Momentum theorem
6. More than one of the above
7. None of the above
8. Cannot be determined

### Commentary:

(5) is the correct response if the rock is thrown radially. Since there
is no angular impulse, there can be no change in angular momentum.
Neither the rock alone, nor the child/disk system changes angular
momentum.

#### Background

Throwing the rock radially, clearly increases the kinetic energy but not
the angular momentum. Consequently, the final angular speed of the disk
and child is the same as the initial speed.

#### Questions to Reveal Student Reasoning

Does the rock have angular momentum (or energy) just before it is
thrown? just after it is thrown?

If energy (angular momentum) is gained, where does it come from?

Changes in angular momentum are caused by a net torque. What torques
act on the system during the process of throwing?

#### Suggestions

Have students relate their answer to this question to item 67.

# A2L Item 067

Goal: Recognize physical conditions under which conservation principles hold.

Source: UMPERG-ctqpe144

A
child is standing at the rim of a rotating disk holding a rock. The
disk rotates without friction. The rock is thrown in the RADIAL
direction at the instant shown. What quantities are conserved during
this process?

1. Only angular momentum is conserved.
2. Only mechanical energy is conserved.
3. Both angular momentum and mechanical energy are conserved.
4. Neither is conserved.
5. cannot be determined.

### Commentary:

(1) is the correct response if the rock is thrown radially. The change
in velocity of the rock and, therefore its change in momentum, is in the
radial direction. The net torque on the system is zero so the angular
momentum cannot change. Some students may be tempted to choose (3) but,
since the rock is thrown via biological processes (as opposed to
mechanical processes), mechanical energy is not conserved.

#### Background

Throwing the rock radially, clearly increases the kinetic energy but not
the angular momentum. This item provides a mechanism for a rich
discussion of the source of the kinetic energy.

#### Questions to Reveal Student Reasoning

Does the rock have angular momentum (or energy) just before it is
thrown? just after it is thrown?

If energy (angular momentum) is gained, where does it come from?

Changes in angular momentum are caused by a net torque. What torques
act on the system during the process of throwing?

#### Suggestions

Have the students do a ‘thought’ experiment by considering a spring
loaded gun mounted on a rotating turntable aimed outward along a radius.
The spring is released firing a small ball outward. This situation
makes it easier for some students to identify the source of additional
kinetic energy. Further, since the force applied is parallel to the
radius, there is no angular impulse and no change in angular momentum in
the system. Have students relate their answer to this question to the
previous one. Also contrast this and the previous one to items 64 and
65.

# A2L Item 066

Goal: Hone the vector nature of velocity.

Source: UMPERG-ctqpe142

A
child is standing at the rim of a rotating disk holding a rock. The
disk rotates without friction. The rock is thrown in the RADIAL
direction at the instant shown, which of the indicated paths most nearly
represents the path of the rock as seen from above the disk?

1. path (1)
2. path (2)
3. path (3)
4. path (4)
5. path (5)
6. cannot be determined

### Commentary:

(4) is the correct path if the rock is thrown radially.

#### Background

Once thrown the components of the velocity of the rock lying in a
horizontal plane are constant so the rock will have a path which is a
straight line.

#### Questions to Reveal Student Reasoning

Identify a coordinate frame. What are the components of the velocity
vector immediately after the rock is thrown?

What is the radial component of the velocity if the rock follows path
(2)?

Is it possible to throw the rock in such a way that the rock follows
path (5)?

#### Suggestions

This item should be compared to 63.