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 074

Goal: Problem solving using momentum conservation.

Source: UMPERG-ctqpe90

On an icy road, an automobile traveling east with speed 50 mph collides
head-on with a sports car of half the mass traveling west with speed 60
mph. If the vehicles remain locked together, the final speed is:

1. 10 mph, West
2. 20 mph, West
3. 30 mph, West
4. 10 mph, East
5. 20 mph, East
6. 30 mph, East
7. The vehicles remain stationary.
8. None of the above
9. Cannot be determined

Commentary:

(8) None of the above. This is a straightforward totally inelastic
collision situation.

Background

Students are frequently bothered by the idea of a totally or perfectly
inelastic collision. They are inclined to think of inelasticity as
imperfection, so the idea of perfect imperfection is distressing.
Consequently the scale shifts and they label collisions when objects
stick together as inelastic, the general collision as elastic, and
collisions conserving kinetic energy as perfectly elastic.

Questions to Reveal Student Reasoning

How fast would the car have to be traveling for the combined vehicles to
remain at rest after the collision?

If the collision was elastic, in which direction would the sports car
travel after the collision?

Suggestions

By relating the general collision problem to that of two masses
colliding with a spring between them, it is possible to get students to
realize that all two body collisions pass through the state with both
objects traveling with the CM velocity. This helps unify the concepts
of elastic, inelastic and perfectly inelastic collisions.

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 065

Goal: Recognize physical conditions under which conservation principles hold.

Source: UMPERG-ctqpe134

A
child is standing at the rim of a rotating disk holding a rock. The
disk rotates without friction. The rock is dropped 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:

(3) is the correct response if the rock is simply dropped. Some
students may fail to include the rock as part of the system after it is
dropped.

Background

Objects traveling in a straight line do have angular momentum with
respect to any origin that is not on the path of the object. The rock
does not cease to have angular momentum with respect to the center of
the disk when it is dropped. Although the angular momentum and energy of
the rock will change as the rock falls, its angular momentum and energy
just after it is dropped are the same as just before.

Questions to Reveal Student Reasoning

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

If energy (angular momentum) is lost, what happens to it?

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

Suggestions

Have students relate their answer to this question to the previous one.

A2L Item 022

Goal: Link energy and kinematic quantities.

Source: UMPERG

Two masses, m and M, are released from rest at a height H above the
ground. Mass m slides down a curved surface while M slides down an
incline as shown. Both surfaces are frictionless and M > m.

Which of the following statements is true?

1. The time it takes for m to reach the end of the surface is longer because the path it takes is longer.
2. The time it takes is the same since both masses are released from the same height.
3. The time it takes for M to reach the end of the incline is less because its horizontal acceleration is larger.
4. The time it takes for m to reach the end of the surface is shorter because it has a larger acceleration initially and therefore builds speed more quickly.
5. The time it takes is the same since both masses have the same displacement.
6. The time it takes is the same because both masses have the same speed at the end.
7. The times cannot be compared without knowing the masses of the blocks.
8. More than one statement above is correct
9. None of the above statements is correct.

Commentary:

(4); even though both blocks arrive at the bottom with the same speed, m
has a larger initial acceleration and attains a larger speed faster than
M, despite having to travel a slightly longer distance. This item helps
to focus attention on identifying those salient characteristics of the
problem that relate to the time it takes the blocks to slide down the
ramps. Some students will cue on the distance traveled, some on the
differing masses of the blocks, some on m picking up speed faster than
M.

Background

The curved surface makes it impossible for students to use either
kinematics or Newton’s Second Law to determine the exact time it takes m
to reach the bottom. Some students may correctly conclude that both
blocks arrive at the bottom with the same speed, and thereby erroneously
conclude that this must mean they arrive at the same time as well.

The curved track case also offers an opportunity to explore whether
students realize that the total work done by the gravitational force
goes into changing the kinetic energy of the block, even with a normal
force present since this normal force does no work on the block.

Questions to Reveal Student Reasoning

What features of the problem determine the time it takes the masses to
reach the bottom?

What’s the same about both blocks if they are released from the same
height? What’s different?

Does traveling a shorter distance always mean less time?

Suggestions

For those who answered (1), ask what would happen to the time it would
take M to reach the bottom if the 45° angle were made more, or less
steep (think of the top vertex of the triangle being on a hinge).
Clearly in the limit where M would drop vertically a distance
SQRT(H2+L2), the time it would take to reach the
other vertex of the hypoteneuse would be shorter than for any angle less
than 90°.