Ratio of work done using pulley

In summary, the conversation discussed using an ideal pulley and massless rope to lift a heavy box, and the magnitude of force and work done in different methods of lifting the box. The ratio of work done in lifting directly and using a pulley is equivalent and therefore has a ratio of 1.
  • #1
negation
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0

Homework Statement



As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. There is no one around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. You pull up on the rope to lift the box.

The Attempt at a Solution



a) Once you have pulled hard enough to start the box moving upward, what is the magnitude F of the upward force you must apply to the rope to start raising the box with constant velocity?
Express the magnitude of the force in terms of m, the mass of the box.

F = 4.9 m?

b) Part B

Consider lifting a box of mass m to a height h using two different methods: lifting the box directly or lifting the box using a pulley (as in the previous part).
What is Wd/Wp, the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?
Express the ratio numerically.

I know that in both cases, the work done by lifting the box directly or by using the pulley are equivalent since in lifting the box, the work done is wd = Fxd = mgh.
In using a pulley, the force is halved but the distance is 2h.
 
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  • #2
negation said:
F = 4.9 m?
Since no units have been specified, you cannot plug in g = 9.8. Just leave it as g.
b) Part B

Consider lifting a box of mass m to a height h using two different methods: lifting the box directly or lifting the box using a pulley (as in the previous part).
What is Wd/Wp, the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?
Express the ratio numerically.

I know that in both cases, the work done by lifting the box directly or by using the pulley are equivalent since in lifting the box, the work done is wd = Fxd = mgh.
In using a pulley, the force is halved but the distance is 2h.
Right, so your answer is?
 
  • #3
haruspex said:
Since no units have been specified, you cannot plug in g = 9.8. Just leave it as g.

Right, so your answer is?

In response to part(a): would mg/2 do fine?

lifting directly:

w = fxh = mgh

using pulley: F + T = mg
2F = mg
F = mg/2
but distance is now 2h
2h(mg/2) = mgh
w = mgh
Since work done are both the same, then the ratio has to be 1?
 
Last edited:
  • #4
negation said:
In response to part(a): would mg/2 do fine?

lifting directly:

w = fxh = mgh

using pulley: F + T = mg
2F = mg
F = mg/2
but distance is now 2h
2h(mg/2) = mgh
w = mgh



Since work done are both the same, then the ratio has to be 1?
Yes, and yes.
 
  • #5
Therefore, the work done is wp = (F/2)(2h) = Fh.
The ratio of work done lifting directly to using a pulley is Wd/Wp = wd/wp = mgh/Fh = mgh/(Fh) = mgh/(Fh) = 1/2

I agree with your reasoning and calculations. The ratio of work done directly to using a pulley is indeed 1/2. This is because, in the case of using a pulley, the force is divided by 2 but the distance is doubled. This results in the same amount of work being done in both cases. This is a fundamental principle of pulleys, where the mechanical advantage is equal to the number of ropes supporting the load. In this case, since there is only one rope supporting the load, the mechanical advantage is 2. This means that the force needed to lift the box is half of the weight of the box. This makes it easier for you to lift the box and reduces the amount of work required.
 

What is the ratio of work done when using a pulley?

The ratio of work done using a pulley is typically 1:1. This means that the amount of work done by the input force (effort) is equal to the amount of work done by the output force (load).

How does the number of pulleys affect the ratio of work done?

The number of pulleys does not affect the ratio of work done. The ratio remains 1:1 regardless of the number of pulleys in the system. However, using multiple pulleys can make it easier to lift heavier loads by distributing the weight among multiple ropes or cables.

Does the weight of the pulley itself affect the ratio of work done?

No, the weight of the pulley does not affect the ratio of work done. The ratio remains 1:1 regardless of the weight of the pulley. However, the weight of the pulley should be taken into consideration when calculating the overall load being lifted.

How does the angle of the rope or cable affect the ratio of work done?

The angle of the rope or cable can affect the ratio of work done. The closer the angle is to 90 degrees, the more efficient the pulley system will be. This is because as the angle decreases, the amount of effort required to lift the load also decreases. However, the ratio of work done will still remain 1:1.

Is there a limit to the amount of weight that can be lifted using a pulley system?

There is no limit to the amount of weight that can be lifted using a pulley system. However, the more pulleys used, the more efficient the system will be. Additionally, the strength and capacity of the rope or cable being used must also be taken into consideration when determining the maximum weight that can be lifted.

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