Two Pulleys and a Weight

Classical Mechanics Level 3

In which arrangement of pulleys given below do we need less force to lift the weight?

Details and Assumptions:

The pulleys are massless.
The friction is absent everywhere.

A B Same force is required in both arrangements

2 solutions

Pranshu Gaba
Feb 18, 2017

In both cases the mass of the block is the same, yet we will see that we require less force to pull the block in case A compared to case B.

In A, the pulley above the block is not fixed. When we pull on the string, the pulley moves up and block rises along with it. If we pull the rope with force $F$ , then the mass feels a force of $2F$ upwards.

In B, if we pull the rope with force $F$ , the mass feels a force of $F$ upwards.

To lift a 10 kg block, it needs to feel a upward force which is slightly greater than its weight, 100 Newtons.

In case A, we need to pull the rope with force greater than 50 Newtons, whereas in case B, we need to pull the rope with force greater than 100 Newtons. Therefore we require to pull the rope with less force in case A.

We are able to lift the heavier mass with a smaller force. Doesn't that mean if both the blocks were to be raised to the same height, we will have to do less work on the block $A$ ?

Rohit Gupta - 4 years, 3 months ago

Because we have to apply half the force in A, it would seem that we can "cheat" and need only half the energy to lift the block. This cannot be true because it would be a violation of the law of conservation of energy.

In case A, we can apply half as much as force than case B. However, if we want to pull both blocks to the same height, we need to pull the rope in A twice the length compared to B. We can see that in A, if pull the rope by length $l$ , the block rises by only $l/2$ . Whereas in B, if we pull the rope by length $l$ , the block rises by length $l$ .

The result is that Force $\times$ distance is the same in both cases, so the amount of energy is required is also same. Law of conservation of energy holds true.

Pranshu Gaba - 4 years, 3 months ago

Yes, at first it appears that we will save a lot of energy if we build a crane with such a pulley system. Although that is not true, such pulley systems will only allow cranes to carry greater loads but not to save any energy.

Rohit Gupta - 4 years, 3 months ago

The masses are equivalent, as given in the problem.

Brad Helland - 4 years, 3 months ago

Yes, same mass can be pulled by smaller force or the same force/machine will be able to carry heavier mass.

Rohit Gupta - 4 years, 3 months ago

Michael Mendrin
Feb 23, 2017

Another way of looking at this is to note that for every foot of rope pulled from the right ceiling pulley, the weight in A rises half a foot while the the weight in B rises a foot. Hence, it takes less force to lift the weight in A, as less work is being done with each foot of rope pulled.

This is a nice way to look at the problem. Work done in this case is the force applied on the rope times length of rope pulled. If the length of rope pulled is kept constant, then work done is proportional to force applied.

Pranshu Gaba - 4 years, 3 months ago

Thank you! This is a perfect explanation for non-physicist!

Joshua Nesseth - 4 years, 3 months ago

Two Pulleys and a Weight

2 solutions

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