Goal: Reason using 2nd law.
Source: UMPERG-ctqpe24
Consider the two situations presented below. T1 is the tension in the string in case A and T2 is the tension in the string in case B.
Which of the following statements is correct?
Goal: Identifying and classifying forces.
Source: UMPERG
Three blocks are stacked as shown below.
How many forces are acting on the bottom block (m3)?
(3); The gravitational force exerted by the earth, the normal force
exerted by the horizontal surface, and the normal force exerted by the
block with mass m2.
Some students have difficulty distinguishing between direct and indirect
interactions. Students may take the view that m1 directly
exerts a force on m3. This view is often verbalized as “the
weight of block m1 is exerted on m3.”
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.
Does m3 exert a force on m1?
What part of m2 interacts with m3? What part of
m1 interacts with m3?
Is weight a force? If so, what object exerts the force?
Can an object interact with another object without touching it? If so,
when? If not, why not?
Is the normal force exerted by m2 on m3 less than,
equal to, or greater than the weight of m2?
If one pushes on both sides of a bathroom scale the scale reading will
change. What does the scale measure. How is the reading related to the
forces exerted on the scale?
If bathroom scales are placed between the blocks, what forces would each
scale measure (assuming that the scales themselves have very little mass
compared to the mass of the blocks)?
Goal: Hone the concept of normal force.
Source: UMPERG
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 normal force on the ball?
Enter (9) if the direction cannot be determined.
(1) By definition the normal force is always perpendicular to the
surface at the point of contact, independent of the motion of the object
and the shape of the surface. The direction of the normal force is away
from the surface and toward the object in contact with it.
When the normal force is introduced to students, a flat surface is used
to illustrate the concept. Flat surfaces are also used in the majority
of problems that students solve. This item extends the context so that
students consider the normal force exerted on an accelerated object
moving on a curved surface.
Those who answer (8) may be thinking that the normal force always
opposes the gravitational force, as when an object is resting on a
horizontal surface.
Students who answer (5) may be indicating the direction of the normal
force exerted on the curved track by the ball.
If a ball were on a flat horizontal surface, what would be the direction
of the normal force? What would be the direction of the normal force if
the ball were rolling across a flat horizontal surface? What would be
the direction of the normal force exerted on a block at rest on an
incline? What would be the direction of the normal force on a ball
rolling down an incline?
What direction(s) are perpendicular to the track at point B?
The direction of the normal force is essentially a matter of definition.
The track exerts a force on the ball. Dividing this force into a
component perpendicular to the surface (called the normal force) and a
component tangential to the surface (called the friction force) is a
choice, which is made because it is useful to do so. Definitions are
difficult to get across to students because there are no demonstrations
one can do to show that the normal force points in a particular
direction. The only thing one can verify is how the definition is
applied by students in a diverse set of contexts.
Goal: Reasoning
Source: UMPERG
Consider the arrangement of pulleys and masses shown below. The masses
of the pulleys are small. Ignore friction.
This system is initially at rest. What will happen if the ring R is
moved to the right?
(3) If the ring is moved to the right the upward force on M is
decreased,so M will accelerate downward. Initially the tension is Mg/2.
When the strings are at an angle the tension is insufficient to support
M.
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?
Does the tension stay the same? …increase? …decrease? After moving
the ring, would I need a smaller or larger mass M to keep the system
from moving?
After students make predictions and discuss their reasoning have
students vote a second time. Then demonstrate what happens.
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?
(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.
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: 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: 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: Interpreting strobe diagrams in conjunction with verbal information.
Source: UMPERG
Below is shown a strobe diagram indicating the position of four objects
at successive time intervals. The objects move from left to right.
Each of the objects shown experiences a constant friction force as they
slide across the floor. Which of the objects definitely experiences a
force in addition to sliding friction?
(8); Object (C) slows down and this could be due to a sliding friction force acting to the left. Given that there is a friction force, the objects moving with constant velocity (B and D) must have an additional force. Likewise (A), which accelerates duting the first part of its motion, must have an additional force to the right.
Commentary:
Answer
(2). The force exerted on each block by the attached string must
balance the weight of the block. Since the blocks all weigh the same
amount, the tension in the in the two strings must be equal.
Background
This item does not require formal knowledge of Newton’s Second Law. It
can be used after students can identify the tension and gravitational
force, provided they appreciate that for static situations the forces
exerted on each object must balance. Try asking students to answer the
question individually and without discussion, giving their initial
reaction. Then ask students to re-answer the question after discussing
it briefly in small groups.
Students commonly think that T2 is greater than
T1 because the rope in situation 2 “supports” two masses.
This incorrect intuition can even exist in students who are capable of
drawing correct free-body diagrams and who know that the “tension” force
exerted on each block must balance the weight of the block. The
coexistence of conflicting intuition and formal knowledge is common
among novices.
Questions to Reveal Student Reasoning
What is tension? How do you measure tension?
For situation (A) consider placing a spring scale between the string and
the block in the vertical region. What force is the spring scale
measuring? For situation (A) consider cutting the string in the middle
of the horizontal region and inserting a spring scale. What force is
this spring scale measuring? How would the readings on the two spring
scales compare?
If spring scales were placed similarly in situation (B), how would their
readings compare to the readings on the spring scales in situation (A)?
Suggestions
Consider the original situations and two variations: (a) In (A)
consider a person holding the string in place of the wall; (b) In (B)
consider a person holding the string in place of the block on the left.
In each situation, what force is being exerted on the string so that the
hanging mass at the other end does not move?
Set up the two situations depicted in the item. Insert spring scales at
appropriate points. Discuss the readings on the scales.