Multiple Choice quiz: two-body problem in an elevator

Diagram for today's problem, modified from
something in (I think) Serway & Vuille

A couple of nights ago, I assigned a two-body problem in an elevator, from (I think) Serway & Vuille.  Two blocks were hanging from an elevator as shown in the picture; the acceleration in the original problem was upward.  On the homework, I asked (among other things):

• Draw a free body diagram for each object.
• Is the tension in the lower rope greater than, less than, or equal to 35 N?
• Calculate the tension in each rope.
• The ropes have a breaking tension of 85 N.  Calculate the maximum acceleration that will cause a rope to break.
• When a rope is observed to break, explain how the elevator was moving.

This problem is one of the best at separating those who are following an appropriate physics problem solving procedure from those who are just trying to plug numbers into some random equation.  The students who used the free body diagrams to write (up forces) - (down forces) = ma got the right answers, and got them quickly. 

On the other hand, the students who didn't carefully write the equations were confused for most of an hour, got the final answers correct because they asked friends for help, but usually earned little credit -- if after collaboration they just wrote "T = ma + 35 N, so T = 40 N" I marked the answer wrong.  Why?  Because I saw no evidence of how they got to that equation, other than listening to a friend without understanding.  Would an English teacher give credit for a one-sentence essay, even if the one sentence is spot-on in its conclusion?  Of course not.  So why on homework should I reward the correct numerical answer when it was essentially derived through magic?

I invited in for extra help the students who didn't follow the correct method.  They now feel much more confident about two-body problems, because they see that all they have to do is write the correct Newton's Second Law equations from the free body diagrams.  But it's still worth a follow up quiz -- either I build significant confidence, or I discover further misconceptions.

Below is today's three-question quiz that I'll give at the opening of class.  (It refers to the diagram above, in which the acceleration is DOWNWARD.  Yeah, I switched the direction of acceleration for the quiz.)  The "distractor" answers in the second question quote some students verbatim.  

Two 3.5 kg blocks hang from ropes in an elevator, as shown above.  The acceleration of the elevator is 1.6 m/s², downward.  While the elevator has this acceleration, the tension in the bottom rope is 29 N.

19. Which of the following best describes how the elevator’s speed is changing?

(A) The elevator is speeding up.

(B)  The elevator is slowing down.

(C)  The elevator is moving at constant speed.

(D) Whether the elevator is speeding up or slowing down cannot be determined.

  

20. Which of the following describes the meaning of an acceleration of 1.60 m/s²?

(A) The elevator gains or loses 1.6 meters per second of speed each second

(B)  The elevator gains or loses 1.6 meters each second

(C)  The elevator travels 1.6 more or fewer meters each second

(D) The elevator travels 1.6 m/s² more or less each second

(E)  The elevator is either speeding up or slowing down by 1.6 meters for every second squared.

  

21.  Now the magnitude of the elevator’s acceleration is doubled to 3.2 m/s², still directed downward.  What is the tension in the bottom rope now?

(A) 41 N

(B)  35 N

(C)  32 N

(D) 24 N

(E)  0 N (i.e. the rope goes slack)

Two-body problem quantitative demonstration

Consider the classic two-body problem, in which a cart on a horizontal low-friction track is attached via rope-and-pulley to a hanging mass.  The cart has mass 340 g, and the hanginging mass is 100 g.  I release the cart, which speeds up.

Is the tension in the rope...
(A) greater than 1 N
(B) less than 1 N
(C) equal to 1 N?

"Equal," say a majority of the students -- because the rope is attached to the 100 g hanging mass, which has a weight of 1 N.

"Less than 1 N," the cleverer ones respond.  The hanging mass has a downward acceleration, because it is moving down and speeding up.  So the down forces must be GREATER THAN the up forces, meaning the tension is less than the 1 N weight.

"Let's see," I say.  The picture shows my cart with a Vernier force probe taped inelegantly on top.  The string is attached to the force probe, is run over a low-friction Pasco pulley, and connected to a 100 g hanging mass.  I tell Vernier's Logger Pro software to collect force probe data.  Before I let go of the cart, the probe reads 1.0 N.  As soon as I let go, the force probe's reading very obviously dips, to something like 0.7 - 0.8 N.  Looks like the cleverer ones were right.

Next, I use free body diagrams and Newton's second law to predict the acceleration of the cart and the tension in the rope -- I get 2.3 m/s² and 0.77 N.  Sure enough, that's just about what the force probe and a motion detector read.  (I get the acceration from the slope of a velocity-time graph, which I make with the vernier motion detector.)

Next question:  Instead of letting the cart go from rest, I give the cart a shove to the right, away from the rope. 

After I let go but as the cart is still moving to the right, is the tension...
(A) greater than 1 N
(B) less than 1 N
(C) equal to 1 N

"Greater than 1 N," say the majority.  "The hanging mass is now moving upward, so the up forces must be greater than the down forces."

"Nonsense," I say.  The mass is slowing down while it moves upward.  Slowing down means that acceleration and velocity are in opposite directions.  Thus, the acceleration must still be in the downward direction, and the tension must still be less than the weight.

In fact, the entire set of free bodies is unchanged from the previous problem -- the force that I pushed with doesn't act once I let go!  So, all calculations are the same, and the acceleration and tension should be unchanged.

Of course, I finish the class by doing the experiment -- sure enough, the tension and acceleration readings are the same as before.

GCJ