# A2L Item 261

Goal: Recognize a lack of information

Source: CT151.2S02-24 Consider the situations at right. Let m < M. Which spring has
the largest spring constant?

1. A
2. B
3. C
4. A and B are equal
5. A and C are equal
6. B and C are equal
7. All of them have the same spring constant
8. Cannot be determined

### Commentary:

#### Answer

(8) The objective of this question is to reveal what students are
assuming about the springs. The reasoning behind any incorrect answer
should be thoroughly discussed.

# A2L Item 183

Goal: Reason with impulse and energy

Source: CT151.2S02-46 Two
blocks are connected to the ends of a spring as shown. Assume that the
mass is proportional to the size of the block. The spring is compressed
(same amount) and released suddenly. In which orientation will the
system achieve the largest height?

1. A
2. B
3. both go to the same height
4. cannot be determined

### Commentary:

#### Answer

(2) This is a very rich problem for reasoning. It IS possible for
students to reason to the correct solution if they consider appropriate
concepts. To help them along suggest the following: Draw free body
diagrams for each of the masses separately. Combine them to get a valid
free body diagram for the system. Such a process reveals that the normal
force is responsible for the impulse causing the system to jump. The
spring force is internal to the system and does not appear on the
system’s free body diagram.

Students can deduce the answer using analogy or experience. Pogo sticks
or even the human body are analogous systems.

# A2L Item 155

Goal: Problem solving

Source: UMPERG-ctqpe84

A mass of 0.5 kg moving along a horizontal frictionless surface
encounters a spring having k = 200 N/m. The mass compresses the spring
by 0.1 meters before reversing its direction. Consider the total time
the mass is in contact with the spring. What is the total impulse
delivered to the mass by the spring?

1. -4 N-s
2. -2 N-s
3. 0 N-s
4. 2 N-s
5. 4 N-s
6. none of the above
7. cannot be determined.

### Commentary:

#### Answer

(2) This problem requires students to put together the concepts
of kinetic and potential energy, and change of momentum. Some may be
tempted to resort to the definition of impulse and try to determine the
force due to the spring.

# A2L Item 149

Goal: Reason with potential energy

Source: UMPERG-ctqpe68 Consider three spring systems having identical springs and
masses. Which of the 3 systems has the largest amount of stored energy?

1. A
2. B
3. C
4. A and C
5. All have the same potential energy
6. Cannot be determined

### Commentary:

#### Answer

(2) Students often forget to include the gravitational potential
energy and also have difficulty selecting a good reference point.

# A2L Item 100

Goal: Reasoning about work and energy.

Source: UMPERG-ctqpe72 Consider the two situations shown above. The springs are identical and
are compressed the same amount, but the masses are different with M > m.
The surfaces they sit on have the same non-zero coefficient of friction.
Both start from rest. Which mass has the largest speed when the spring
reaches its relaxed length?

1. m
2. M
3. Both have the same speed.

### Commentary:

#### Answer

(1) Friction is only a confounding element. The lighter mass will have
the greater speed whether or not there is friction.

Students may correctly reason that the friction force will be less on m
and less of the potential energy stored in the spring will be dissipated
as the spring returns to its relaxed length. While true this is not
relevant for the question.

This is an instance where it is important to elicit student reasoning.
It is a case where students can use wrong reasoning to get the correct
answer.

# A2L Item 093

Goal: Problem solving

Source: UMPERG-ctqpe88 Two blocks, M=2m, sit on a horizontal frictionless surface with a
compressed massless spring between them. After the spring is released
M has velocity v. The total energy initially stored in the spring was:

1. mv2
2. 2mv2
3. 3mv2
4. 4mv2
5. 5mv2
6. None of the above.
7. Cannot be determined

### Commentary:

#### Answer

(3) The big mass has kinetic energy mv2 and the small mass
has energy 2mv2. Some students may answer (6) because they
have confused M and m. It is important to determine the reasons that
any student might select (7). They might be unwilling to assume that
the system is initially at rest. Students taking this perspective
should not be disconfirmed but congratulated for making a critical
interpretation of the wording.

# A2L Item 070

Goal: Recognizing the presence of forces.

Source: UMPERG A block having mass m moves along an incline having friction as shown in
the diagram above. As the block moves a small distance along the
incline, how many forces act on the block?

1. One force
2. Two forces
3. Three forces
4. Four forces
5. Five forces
6. Six forces
7. Seven forces
8. None of the above
9. Cannot be determined

### Commentary:

#### Answer

(5) Five forces act on the block: gravitation, rope, spring, kinetic
friction (because you are told the block moves), and normal due to the
incline. Many student errors are due to the failure to identify all of
the forces acting on a body.

#### Background

It is helpful to classify forces into action-at-a-distance forces, such
as gravity and electromagnetism, and contact forces. Students can then
employ a strategy for identifying all the forces since every object
touching a body will give rise to a force. The only exceptions are the
fundamental forces, which is an easily exhausted list.

#### Questions to Reveal Student Reasoning

Does it matter if the block is moving up the plane or down? If the block
is at rest, how many forces MUST be acting on the block? How many forces
may be acting but you can’t be sure?

#### Suggestions

Set up some situations with blocks, springs and ropes and let students
practice identifying all the forces. This is a good activity to do in
conjunction with drawing free body diagrams.

# A2L Item 062

Goal: Interlink several dynamical concepts and associate with a physical process.

Source: UMPERG-ctqpe98

Mass M1 is traveling along a smooth horizontal surface and
collides with a mass M2 (stationary) which has a spring
attached as shown below. The spring between the blocks is most compressed when

1. all the energy in the system is potential energy stored in the spring.
2. the net momentum of the system is zero.
3. the velocity of the center of mass has its smallest value.
4. mass M1 is no longer delivering an impulse to Mass M2.
5. the only kinetic energy in the system is that of the center of mass.
6. none of the above
7. cannot be determined

### Commentary:

#### Answer

5: When the spring is maximally compressed, both masses have the same
velocity which is the velocity of the center of mass.

#### Background

The total energy of an isolated system can be decomposed into three
categories; kinetic energy associated with center of mass motion,
kinetic energy of bodies in the center of mass coordinate frame and
potential energy associated with the interaction of bodies comprising
the system.

#### Questions to Reveal Student Reasoning

Is M2 ever traveling faster than M1?

Do the masses ever have the same velocity?

How would you find Pcm, the momentum of the center of
mass?

How is the kinetic energy of the center of mass related to its
momentum?

#### Suggestions A
sketch of the velocities of the two masses over time would look
something like the graph at the right. [Note that the velocity of
M1 can be negative after the collision if it is less massive
than M2.] Such a graph helps make the relationships between
Vcm, the relative velocity of the masses and the spring
compression clear.

# A2L Item 052

Goal: Reasoning and comparing the sizes of forces.

Source: UMPERGA block attached to the end of a spring is hanging at rest from the

A block attached to the end of a spring is hanging at rest from the
ceiling as shown at the left below. After the block is pulled down and
released it moves up and down for an extended period of time. The
motion during one cycle is shown in the graph at right below. Several points are indicated on the graph. At which point is the spring
force exerted on the block the greatest?

1. Point A
2. Point B
3. Point C
4. Point D
5. Points B and D
6. Points A and C
7. The spring force is always the same
8. None of the above
9. Cannot be determined

### Commentary:

#### Answers

(4). The spring force is largest at the position where it is compressed
or stretched the most relative to its natural length. The spring is
already stretched when it is at a height H because there must be an
upward spring force to balance the gravitational force on the block. As
the height of the block is decreased the spring is stretched further.
As the height of the block is increased the spring is stretched less –
if raised enough the spring would start to compress.

#### Background

Many students will attempt to apply the spring force law without real
understanding. This problem requires students to understand the
physical situation and to interpret graphical information about the
height of the block to reason out an answer.

#### Questions to Reveal Student Reasoning

What is the force law for a spring? How does the spring force compare
to the weight of the block? At what points is the spring stretched? …
compressed?

#### Suggestions

Demonstrate with a spring that a vertical spring stretches when a weight
is attached. Show that as the weight moves up and down that the spring
need never get back to its natural length (i.e., it is always stretched)

Draw free-body diagrams, especially for points B and D.

# A2L Item 037

Goal: Translate a verbal description of physical motion to graph of force.

Source: UMPERG

A block is dropped onto a vertical spring. Which net force vs. time
graph best represents the net force on the block as a function of time?
Consider only the motion of the block from the time it is dropped until
it first comes to rest. ### Commentary:

#### Answer

(4); Some students may select (5) confusing the equilibrium point with the point where the block comes to rest. Students selecting (2) are ignoring gravity after the block hits the spring.