# 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 153

Goal: Hone the concept of impulse

Source: UMPERG-ctqpe80

A MOVING car collides with a STATIONARY truck. Which of the following
statements is true about the magnitudes of the impulse on each due to
the other?

1. The impulse on the car is larger than the impulse on the truck.
2. The impulse on the truck is larger than the impulse on the car.
3. The magnitudes of the two impulses are equal.
4. Answer depends upon circumstances of the collision.

### Commentary:

#### Answer

(3) The 3rd law requires that the impulses be equal. Even
students who understand the 3rd law have difficulty realizing that the
magnitude of the impulse on two interacting bodies is the same. Many
students, however, do not understand impulse enough to recognize the
association. Others do not read the problem carefully enough and answer
with regard to ALL the forces, not just the one due to the other
vehicle.

# A2L Item 151

Goal: Reason with kinematics

Source: UMPERG-ctqpe75

Two identical steel balls are released from rest from the same height,
and travel along tracks as shown and labeled below.

Which ball reaches the end of its track first?

1. ball on track A
2. ball on track B
3. they reach the end at the same time
4. not enough information

### Commentary:

#### Answer

(2) The ball on track B accelerates down the second slope. A
component of this acceleration is in the x-direction. This means that
the x component of ball B’s velocity is never smaller than that of ball
A. Since the tracks have the same x-dimension, ball B gets there first.

A large majority of students choose answer C incorrectly thinking that
since the balls return to the same height, they have the same speed and,
therefore, arrive at the same time.

# A2L Item 152

Goal: Reason with kinematics

Source: UMPERG-ctqpe76

Two identical steel balls are released from rest from the same height,
and travel along tracks as shown and labeled below.

Which reaches the end of its track first?

1. Ball on track A
2. Ball on track B
3. They reach the end at the same time
4. Not enough information

### Commentary:

#### Answer

(2) The ball on track B accelerates down the second slope. A component
of this acceleration is in the x-direction. This means that the x
component of ball B’s velocity is never smaller than that of ball A.
Since the tracks have the same x-dimension, ball B gets there first.

A large majority of students choose answer C incorrectly thinking that
since the balls return to the same height, they have the same speed and,
therefore, arrive at the same time.

# A2L Item 133

Goal: Reasoning with circular motion

Source: UMPERG-ctqpe37

A small ball is released from rest at position A and rolls down a
vertical circular track under the influence of gravity.

When
the ball reaches position B, which of the indicated directions most
nearly corresponds to the direction of the ball’s acceleration?

Enter (9) if the direction cannot be determined.

### Commentary:

#### Answer

(2) At position B the acceleration has a tangential component and
a radial component. Both components can be determined at position B.
Worked out carefully one gets 18 degrees above position #2. It is common
for students to neglect one component or the other.

# A2L Item 131

Goal: Reasoning with dynamics

Source: UMPERG-ctqpe30

A
block of mass m, when placed on a rough inclined plane and moved, moves
down the plane with constant speed. If a block of mass 2m were placed
on the same incline and moved, it would …

1. return to rest.
2. accelerate until the speed is half.
3. move with some constant speed.
4. None of the above.
5. Cannot be determined

### Commentary:

#### Answer

The block will have the same motion. Both the gravitational force
and the friction force scale with the mass so there is no net force in
either case.

# 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.

# A2L Item 081

Goal: Develop a strategic approach to problem solving.

Source: UMPERG

A bug walks on a rotating disk. Given: Mbug,
Idisk, Rdisk, and ωo when the bug
is at r1. The bug crawls to r2. Find
ωfinal for the system.

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

1. Kinematics only
2. F = ma or Newton’s laws
3. Work-Energy theorem
4. Impulse-Momentum theorem
5. Angular Impulse-Angular Momentum theorem
6. Not enough information given

### Commentary:

#### Answer

(5) This problem helps students develop a principle-based approach to
problems. Many students may think correctly that this is a conservation
of angular momentum problem and not recognize that the general principle
is the angular impulse – angular momentum theorem.

# 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 051

Goal: Reasoning about the vector nature of force.

Source: UMPERG

A rock sits on a hillside. The slope of the hillside is inclined to the
horizontal at approximately 30°.

Which of the forces exerted on the rock is smallest in magnitude?

1. Friction force due to the ground
2. Gravitational force due to the Earth
3. Normal force due to the ground
4. The Friction, Gravitational and Normal force are equal in magnitude
5. Cannot be determined
6. None of the above

### Commentary:

#### Answers

(1). The smallest force is the friction force. The normal force
balances the component of gravity perpendicular to the hillside and the
friction force balance the component of gravity parallel to the
hillside. Since the hillside has a slope of 30°, the tangential
component of gravity is smaller.

#### Background

Many students fail to perceive that the static situation implies a
relationship between the forces. They may think that one requires
information such as the mass, and coefficient of friction to compute the
forces before the forces can be compared.

#### Questions to Reveal Student Reasoning

Can you determine the gravitational force? …normal force? …
friction force? Are there any relationships between these forces? If
so, what are they? Why doesn’t the rock slide down the hill?

#### Suggestions

Draw a free-body diagram.

Set up a demonstration using a block on a plane with adjustable angle.