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Name: H. K.
Status: educator
Grade: 6-8
Location: Northern Mariannas
Country: N/A
Date: 1/3/2006

Students were given several possible answers on a multiple choice test, asking, "What method requires the most force to move the block."

The answers were drawings. The first showed a person sliding a box across a floor. The second showed a person lifting the box using a single pulley with a Mechanical advantage of 1. All they were doing was using the pulley to change the direction of the force applied to the box. I have tried to recreate the pictures:

             |         | ------>
 Box pulled horizontally.

    /  __|__     /\
  /    |       |     |
       |____|     |
  Box lifted by pulley

        |                       _____
        |                       |       |
       V __________ __|____|

Box on lever, left side pushed down I wrote to the generators of the test that I felt the answer to the question: "In which situation will the person use the greatest force?" was not clearly one of the choices. No friction or masses were stipulated. I emailed them and I was given the following response.

"We know from Newton's third law that the magnitude of the force of the plane on the box is the same as the magnitude of the force of the box on the plane. It is the weight of the box that is the source of its force on the plane, but not the entire weight. If the plane were at an angle of 90 degrees, the force of the box on the plane would be zero since the weight vector would be parallel to the plane. If the plane were at an angle of zero degrees, the entire weight of the boxes would be pushing on the plane. In fact the force of the box on the plane is the weight of the box times the cosine of the plane's angle.

The cosine of zero degrees is one, thus the force needed to push the box on the flat frictionless surface is equal to the mass of the box. Since no plane is frictionless, the force needed to move the box is equal to the mass of the box plus the friction of the box on the surface. While the pulley also introduces friction, it is small compared to the sliding friction of the box."

My question is this given only the drawings and no mention of the magnitude of friction in any of the situations, which situation -- the person using the pulley (with Mechanical advantage of one) to lift the box or the person pushing the box-- is truly the correct answer? Can it even be determined?

If you are saying this is a poor exam question, I would agree. The nature of the pulley and the nature of the surface on which the box is sliding would be significant factors. Seeing as how they are not defined, it is difficult to have a definitive answer. Under typical conditions, I would expect the box sliding on the surface to require the greatest force... but that is based on assumptions that might or might not be true. I am on your side.

A discussion of this question would be great in your class, but to determine a grade based on the it would not be fair.

Larry Krengel

Dear H.K.,

I think this is a very bad question, but the reply to your email is even much worse!

My answer to the question would be b) (the second answer), but a) or c) could also be the correct answer. If the floor is frictionless, the force needed to move the box in a) (the first answer) can be infinitesimal up to as large as you like depending on the acceleration. The force in c) depends on where the fulcrum is and could be infinitesimal (if the box is placed an infinitesimal distance from the fulcrum) all the way up to an infinite force (if the force is applied an infinitesimal distance from the fulcrum). The force in b) is given by m(a+g), where a is the upward acceleration of the box. If the box is accelerating upward, the force must equal (if a=0) or exceed the weight of the box.

If I had to make a choice, I would choose b), since the force must in that case at least equal the weight of the box. In both a) and c) the force can be less than the weight of the box and could even be infinitesimal. It could also be larger than the weight of the box, although for a) you might have to glue the box to the surface to get a coefficient of friction larger than 1.

I must comment on the response you received from the generators of the test, which I find incompetent. The person who generated that response seems to know no physics and the explanation is hardly relevant. Most of the answer has to do with inclined planes, which have nothing to do with the question!

The person also does not realize that it is very easy to move a box on a frictionless surface! But since this person thinks that force and mass are the same ("the force needed to move the box is equal to the mass of the box plus..."), I am not surprised. And how does the person know that the pulley friction is small compared to the sliding friction of the box? I could easily produce a pulley that has LOTS of friction (rusted so it does not rotate, perhaps) as well as a surface which would have almost no friction (with oil, perhaps).

If you would like further comments, please let me know. I have been holding myself back (though that may not be obvious) and would be happy to let out some of my stronger feelings!

Best, Dick Plano, Professor of Physics emeritus, Rutgers University

The response you received equates force and mass. These are not equivalent quantities but are related by the equation f=ma where f => force, m=> mass, and a=> acceleration. If we assume that we are talking about moving the same box in all cases the masses are equal in all cases. But that is the force DOWNWARD. In the case of pushing the box sideways you do not have to overcome this force, only the static and sliding frictional forces. To help envision this consider a case where the frictional forces are minimized. A good example would be pushing a car on a level parking lot surface. No one is going to be able to pick up a car but most adults will be able to push a car under ideal conditions (i.e., minimal friction). It takes MUCH less force to push (or pull) a heavy box on a near frictionless surface than it does to lift it.

Other comments:

The force to move the box in a vertical direction must overcome the force downward, where 'a' is the acceleration due to gravity. This is the minimum force required to move it with a single pulley (assuming no friction).

Use of a lever will reduce the force required as long as the box is closer to the fulcrum than the applied force downward.

The third choice, that of sliding the box, is the more difficult case to analyze. If no stipulation of the friction is made then you really do not have enough information to determine the amount of force required. However, that stipulation may have been made in the instructions for the test overall (e.g., 'Assume frictionless surfaces except where specified') or it may be implicit from the level of coverage that the students are assumed to have had (frictionless surfaces would be an introductory level). I would assume a frictionless surface. In this case the force required will be proportional to the acceleration achieved. If you want a very small acceleration only a very small force is required.

A (near) frictionless surface can be achieved in many ways

- add wheels to the box

- air flotation (a vacuum cleaner output can provide enough of an air

cushion to make it easy to push heavy objects around)

- water flotation (put the box on a barge and push the barge

- put the box on the DuPont's Teflon (TM) glides sold for moving furniture

Requiring less force to push a box across a surface than to lift the box is also very real-world (i.e., not just frictionless surfaces, or objects with wheels). For example, I can push a 4-drawer file cabinet filled with documents across the floor but I could never lift it (with a single pulley system).

Greg Bradburn

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