Goal: Hone the vector nature of force.
Source: UMPERG
Three picture frames having the same mass are each hung from a wall
using two pieces of string.
For which situation is the tension in the two strings the greatest?
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?
(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.
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.
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)?
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.
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: 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 ?
Commentary:
Answer
(3) The tension will be largest on the wires that are
most nearly horizontal. The minimum tension is in the vertical wires,
and each wire has a tension equal to half the weight of the picture.
Some students may think that the tension is the same no matter how the
wires are arranged. One way to convince them that this is not the case
is to have two students support a heavy object by pulling on ropes
attached to the object. As they move apart they easily perceive the
need to pull harder.