## Monday, March 29, 2010

### Pulleys and Mechanical Advantage

One of the problems on an FRQ in my course that gives students more trouble than I would expect is the following question:

The first question that gets asked is "What is the mass of the bucket and the sand?"

It usually isn't dealing with the component of the weight on the ramp and the friction that cause students their problem, it is dealing with the pulley system. There are two different approaches that I feel can help students understand how the pulleys are working, if they didn't get to play with and talk about pulleys in junior high. I am going to assume we are in a steady state condition.

The first is to think in terms of forces. In figure B, since a rope transmits the same force throughout it's length, as it passes over the pulley, the pulley is doing nothing but changing the direction of the force, T on the left of Figure B is equal to T on the right of Figure B.

The second is to think in terms of the distance things are moving. In a simple machine, which a pulley is, if you can make the input distance your input force works through be twice the output distance the object you are doing work on moves, you have a mechanical advantage of 2. In figure B, if you have a meter of rope come out the left side of the pulley, you also have a meter of rope that went into the right side of the pulley. That is a 1 to 1 ratio, and therefore the mechanical advantage of the pulley in Figure B is 1.

Applying the force argument to the pulley in Figure C, we see that we have two forces each of strength T pulling up, and they counterbalance the single force Ws acting down. We could write an equation about that 2 T = Ws . We can write an equation about Figure B also, T = T, but that doesn't do us much good.

Applying the distance argument to Figure C, if you think about it you can see that because the rope on the left side is fixed, that is what the horizontal bar on the top of the rope means, for every 1 meter of rope we pull through to the right side of the pulley, the weight hanging on the pulley only goes up by 1/2 of a meter. This may not be completely obvious, so visualize a specific situations to think about it. In moving from the arrangement in Figure D below, to figure E, we have dropped the center of the pulley (and therefore the suspended mass), a distance of 1 meter. How much rope did it take to do that? Clearly we added 1 meter of rope on the left of the pulley in Figure E and also 1 meter of rope on the right of the pulley in Figure E. So we have to let 2 meters of rope out to get 1 meter of movement, this is a mechanical advantage of 2. Thus the tension in the rope is 1/2 the weight of the sand bucket, or T = (1/2) Ws, and 2T = Ws.

In figure A, we have two forces trying to keep the block from moving up the plane, the component of gravity down the plane which I will call Fx and the frictional force. Fx = Wb*Sin(30), since that force would be zero if the angle were zero, and it would be the full weight of the block if the angle were 90 degrees. The frictional force is the coefficient of static friction times the component of the weight of the block creating the normal force between the plane and the block of f, where

We can now write our second main equation stating that the forces acting along the x-axis (taken to be along the plane) for the block in Figure A:

Do we have enough equations to solve for the weight of the sand? We'll look at this more next time. Meanwhile see if you know what the following item is. The small photo on the right is a close up of the item being pointed to.

Content related to this blog posting can be found at HippoCampus under Pulleys.