Goal: Reasoning qualitatively.
Source: UMPERG
Consider the arrangement of pulleys and masses shown below. The masses
of the pulleys are small. Ignore friction.
For what relationship of the masses would the masses remain at rest?
Goal: Understanding action-reaction forces.
Source: UMPERG
A hammer strikes a nail driving it into a piece of wood. Which statement
below is true about the forces exerted during the impact?
(3) The forces are the same size (according to Newton’s third law).
Students’ natural inclination in situations like this is that a moving
object is a more active agent and therefore exerts a larger force, while
a stationary object is the more passive agent and exerts a smaller
force. Students also look at effects: the object that has the largest
change in motion has experienced the largest force.
How can you determine which object experiences the larger force? What
are some of the clues? Do we have any way to relate the effects we
observe to the size of the forces each object experiences?
Newton’s third law, while easily memorized as a principle, is hard to
develop as an intuition and to employ in reasoning about situations.
There is no single experience that can help. One needs to revisit the
third law often in many different contexts.
There are many situations one can use with students. Try a moving block
with a spring colliding with a wall (or another block that is
stationary). In this situation one can use the spring law to help
relate the forces.
Goal: Reasoning.
Source: UMPERG-ctqpe28
A block is on a horizontal surface. When the block is pulled by a rope under
tension T, the block moves with constant speed. If the same tension were
applied to a smaller block made of the same material and at rest on the
same surface, the block would:
(5); in the first case, the net force is 0, so T=μkMg. In
the second case, the static friction force must be overcome for m to
move. Since μs>μk, but m<M, it cannot
be determined if μsmg is smaller or larger than T.
This item requires that students combine knowledge from different
topics: Static Friction, Kinetic Friction, and Newton’s Second Law.
Students have to deduce information (e.g., in the first situation
students must deduce that the kinetic friction force is balanced by the
tension force to give a net force of 0). Students must also know that,
since the static friction coefficient is larger than the kinetic
friction coefficient, the maximum static friction force is larger than
the kinetic friction force. Finally, students must be able to reason
about compensating quantities-in this case, although m goes down, μ
goes up, so the product of m, μ, and g may, or may not, be larger
than T. The relationship between students’ answers and their
assumptions should be the focus of the class discussion, not the
correctness of any particular answer.
Why does the block of mass M move with constant speed? If the block of
mass M were at rest would the tension force cause it to move?
What quantities affect the size of the friction force?
What determines whether the block of mass m will move?
Ask students to consider the limiting case where m is less than, but
almost equal to M. What would happen if m were pulled with tension T.
Students should be able to reason (perhaps with some coaching) that m
will remain stationary since the maximum static friction force is larger
than T.
Then ask them to consider the limiting case where m is much less than M.
What would happen if m were pulled with tension T. Students should be
able to reason that m will accelerate.
Finally, ask what happens “in between” these two limiting cases.
Goal: Reason qualitatively. Consider alternate solution paths.
Source: UMPERG
Two blocks, M2 > M1, having the same speed move
from a frictionless surface onto a surface having friction coefficient
μk as shown below.
Which block stops in the shorter time?
(3); both blocks have the same acceleration and the same initial
velocity, so they must stop in the same length of time.
This problem can be reasoned through without the use of equations.
However, the problem can be solved easy enough algebraically. The item
provides an opportunity for students to reflect on different approaches
for solving problems.
Which block experiences the largest net force?
Which block experiences the largest acceleration?
What determines which block stops first?
Ask students to consider the following questions, and to determine if
their answer to the problem is inconsistent with their answers to these
questions:
If two blocks enter the rough region side by side and have the same
mass, which one will stop first?
If the blocks are connected by a rope, will the time it takes for the
blocks to stop change? Would the time it takes to stop change if the
blocks were glued together?
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.
(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.
Goal: Hone the vector nature of force and how the net force can be determined from the 2nd law.
Source: UMPERG
A soccer ball rolls across the road and down a hill as shown below.
Which of the following sketches of Fy vs. t represents the net vertical
force on the ball as a function of time?
(4); on the horizontal partion there is no net force, and therefore no Fy. On the slope the net force in the y direction is due to gravity and a component of the normal force. Alternatively, since the ball accelerates along the slope, the net force must be parallel to the slope and have a component in the vertical direction.
Goal: Hone the vector nature of force and interrelate model and procedure forces.
Source: UMPERG
A marble rolls on to a piece of felt and slows down.
Indicate the direction that most nearly corresponds to the direction of
the force that the marble exerts on the felt. If none of the directions
are appropriate, or if the answer cannot be determined, respond (9).
(3) The force the felt exerts on the marble is up (normal force) and to
the left (friction force). Newton’s third law tells us that the force
the marble exerts on the felt must be down and to the right. Students
may focus on the normal force alone (4) or the friction force alone (2).
These are not two forces, but the components of a single force.
Students also find it difficult to extract some information from the
dynamical statement “slows down” and integrate this with the familiar
normal force.
This presents an interesting twist to students. The friction force is
usually formulated in terms of a moving object and a fixed surface.
Students may not know for sure whether there is a friction force on the
felt – the felt is not moving. The analysis on the marble is reasonably
straightforward. Newton’s third law can be used to determine the force
on the felt if the force cannot be determined from the situation
directly.
Question students about how they got their answer. Did they use the
force laws that they learned previously? Did they use Newton’s second
or third laws?
Instead of a marble consider a sliding block and see if students think
differently – some students will have difficulty thinking about friction
with a rolling object.
Goal: Reasoning with 2nd law and honing of the concept of tension.
Source: UMPERG
Consider the three cases presented below. Assume the friction force
between the table and block in situations (B) and (C) can be ignored.
Which of the following statements about the tensions in the strings is
true?
(4) By applying Newton’s second law to the hanging block one obtains a
relationship between the tension in the string and the acceleration of
the hanging block: The larger the acceleration the smaller the tension
force. The acceleration is determined by the total mass of the system.
This is a good problem for challenging students to reason without
resorting to writing down a lot of equations. As one of the procedure
forces (tension, normal, static friction), the value of tension requires
application of the 2nd law.
What is the tension in situation (A)? Explain. Is the tension equal to
the weight in situation (B)? (If some students think so explore what
the net force is on the hanging mass, which will lead to a net force of
O, and a contradiction since this implies O acceleration.) Of systems
(A) and (B), which has the larger acceleration?
Ask students to consider limiting cases. What if the string was not
attached to a block on the table (or if the block had almost no mass)?
What would happen if the block on the table had a very large mass ?
Goal: Analyze the role of internal and external forces and the difference between static and kinetic friction.
Source: UMPERG
A person sits in an office chair with small wheels that swivel. The
person claims she can move the chair across the room without touching
anything but the chair, simply by kicking her legs outward. This claim:
Is consistent with Newton’s laws and can be done.
Is consistent with Newton’s laws but cannot be done.
Is not consistent with Newton’s laws and, therefore, cannot be done.
Is not consistent with Newton’s laws but nevertheless can be done.
It is not possible to determine the correctness of the claim.
(1); the process is possible because sudden impulse due to internal
forces can exceed the static friction limit. By rapidly extending legs,
alternated by slow retraction, the chair can be moved. Students are
often aware of this but find it difficult to explain in terms of forces
and dynamics.
Goal: Reasoning using the 2nd law.
Source: UMPERG
A tow truck (2,000kg) pushing a car (1000kg) experiences an average
friction force of 13,000N while accelerating from rest to a final
velocity of 36 mi/hr (16 m/s). The air and the road exert an average
resistive force of 1,000N on the car. What force does the car exert on
the tow truck?
(4) The net force on the car and tow truck is 12,000N (13,000N –
1,000N). The acceleration is 4m/s2. The magnitude of the force between
the two vehicles is 5,000N.
Answers are not as important as approach. What did students do to
understand the physical situation? Did they draw pictures. Did they
draw a free-body diagram.
Ask a couple of students to describe how they approached the problem.
Ask them to describe the steps they took without getting into
mathematical details. For example, did they draw a free-body diagram?
What forces did they consider? What system did they analyze?
After a couple of descriptions of how to approach solving the problem,
work through the problem with help from the class.
Commentary:
Answer
(5); This problem can be reasoned although it is easy enough to solve algebraically. The problem is useful for demonstrating the value of free body diagrams for reasoning.