# 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:

1. 6 rad/sec
2. 8 rad/sec
3. 12 rad/sec
4. 15 rad/sec
5. 20 rad/sec
6. Cannot be determined

### Commentary:

#### Answer

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

# A2L Item 160

Goal: Developing a strategic approach to problem solving

Source: CT151.2S02-44 spinoff

A cart
of mass 2m collides and sticks to a cart of mass m that is initially at
rest. The combination of the two then moves together. Which of the
following principles would be the most efficient way to find the final
speed of the combination?

1. The Work/Energy Theorem
2. Newton’s Laws and the basic equations of motion
3. Conservation of Energy
4. Conservation of Momentum
5. Conservation of Angular Momentum
6. A different principle entirely.
7. Two (or more) of them would be equally efficient.
8. Need more information.

### Commentary:

#### Answer

(4) It is valuable to always associate conservation of momentum
with the third law. In addition, it is worthwhile to distinguish totally
inelastic collisions from typical inelastic collisions.

# A2L Item 159

Goal: Explore momentum concepts

Source: CT151.2S02-44

A cart
of mass 2m collides and sticks to a cart of mass m that is initially at
rest. What is the speed of the combination after the collision?

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

### Commentary:

#### Answer

(2) This question is useful for probing pre-existing ideas about
momentum, and also for distinguishing momentum ideas from kinetic
energy. It should be used just after, or even before, covering momentum
concepts.

Usually students starting momentum already have had some energy, and
kinetic energy in particular. If appropriate, they can be asked if
energy is lost in the collision. Is more or less energy lost if the
carts do not stick together?

# A2L Item 145

Goal: Reasoning with energy

Source: UMPERG-ctqpe64

Two
masses, M > m, travel down the surfaces shown. Both surfaces are
frictionless. Which mass has the largest speed at the bottom?

1. m
2. M
3. Both have the same speed
4. Cannot be determined

### Commentary:

#### Answer

(3) By energy considerations, both would have the same speed.
Students frequently get confused about the mass, thinking that the
larger mass has the greatest potential energy change and therefore has
the greatest speed.

# A2L Item 144

Goal: Reasoning with kinematics

Source: UMPERG-ctqpe63

Two
masses, M > m, travel down the surfaces shown. Both surfaces are
frictionless. Which mass has the largest average speed during
their motion?

1. m
2. M
3. Both have the same average speed
4. Cannot be determined

### Commentary:

#### Answer

(1) This problem is intended to promote discussion of average
speed. Both masses have the same speed at the bottom. Mass m has a
larger acceleration in the beginning because the circular track is
vertical at the outset. Although the angle of the incline is not
specified, the angle is irrelevant. All inclines will have the same
average speed. A simple graph of the speed of each mass versus time
shows that m will have the larger average speed.

# A2L Item 143

Goal: Reasoning with energy

Source: UMPERG-ctqpe62

Two
identical blocks fall a distance H. One falls directly down, the other
slides down a frictionless incline. Which has the largest speed at the
bottom?

1. The one falling vertically
2. The one on the incline
3. Both have the same speed
4. Cannot be determined

### Commentary:

#### Answer

(3) The only force doing work is gravity and both block undergo
the same vertical displacement.

# A2L Item 138

Goal: Problem solving with momentum

Source: UMPERG-ctqpe46

A
block slides along a frictionless surface and onto a slab with a rough
surface. The plot on the right shows the velocity of the blue slab as a
function of time. The slab has mass of 4kg and the block has mass of
2kg. If the block remains on top of the slab, what was its initial
speed?

1. 2 m/s
2. 4 m/s
3. 6 m/s
4. 8 m/s
5. 12 m/s
6. None of the above
7. Cannot be determined

### Commentary:

#### Answer

(6) The block initially has velocity 3 m/s. This problem is
difficult for students. They generally have difficulty obtaining
relevant information from a diagram. In this case they must use the plot
to tell that the final velocity is 1 m/s.

# A2L Item 132

Goal: Link energy conservation and electromagnetism

Source: 283-421 Change of total energy

A
uniform volume distribution of charge has radius R and total charge Q.
A point charge -q is released from rest at point b, which is a distance
3R from the center of the distribution. When the point charge reaches a,
which of the following is true regarding the total energy, E?

1. Ea = -Eb
2. Ea = -2Eb/3
3. Ea = -3Eb/2
4. Ea = -9Eb/4
5. Ea = Eb
6. Ea = 2Eb/3
7. Ea = 3Eb/2
8. Ea = 9Eb/4
9. None of the above
10. Cannot be determined

### Commentary:

#### Answer

(5) Students often forget to include the kinetic energy,
especially after a lot of discussion of potential energy. Many will
simply misinterpret the energy to mean potential energy. Teasing apart
these issues is important.

# 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:

#### Answer

(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 085

Goal: Reasoning and recognizing the implications of momentum conservation.

Source: UMPERG

For ANY collision between two objects there is a time when both of the
objects are traveling with the velocity of the center of mass.

(Assume no external forces act on either object.)

1. True
2. False
3. Depends upon the details of the collision

### Commentary:

#### Answer

(2) This statement is false despite the fact that it is true for just
about all of the instances of collision that students see. In a
perfectly inelastic collision it is certainly true that both bodies have
the velocity of the center of mass after the collision. In a general one
dimensional collision with only spring forces it is also true. For the
statement to be true about a specific collision, there must be a time
when the relative velocity of the two objects is zero. The statement is
clearly false in general for two-dimensional collisons. As an example of
a one-dimensional collision for which the statement is false, consider a
bullet that passes through a block of wood initially at rest. The bullet
slows down and the block speeds up but they never have the same
velocity.