Pulling a chain

1. Aug 13, 2014

Yashbhatt

1. The problem statement, all variables and given/known data

A 2 m long chain of mass 4 kg rests on a table such that 60 cm of it hangs vertically downwards from the table. If the chain has uniform mass distribution, calculate the work done in pulling the entire chain upwards. Ignore the frictional force.

2. Relevant equations

W = U = mgh, W = Fdcosθ

3. The attempt at a solution

First, I calculated the mass of the 60 cm section of the chain using ratios. Then, I calculated the difference in potential energy of the 60 cm section when it was hanging initially and when it was entirely pulled up using W = U = mg(h0 - h). I got the value as 7.2 J. But the answer was 3.6 N. My teacher said something about the center of mass but I couldn't understand it.

2. Aug 13, 2014

olivermsun

You aren't pulling the entire 60 cm section upward by 60 cm. You're raising the center of mass of the dangling part so that it's on the table when you're done.

3. Aug 13, 2014

Yashbhatt

But won't we still have to apply force after the center of mass is on the table(which in this case would be at the midpoint of the chain) to pull the rest of the chain upwards is on the table?

P.S. The question says "...pulling the entire chain upwards."

4. Aug 13, 2014

Nathanael

Only the very end of the chain moves up by 60cm. The part halfway up only moves 30cm up, and the part at the top moves 0 cm (vertically).
(etc. etc. etc. for every point on the chain)

So the average height that the chain moves up is only 30cm

The way to solve this is to look at the center of mass of the 60cm part of the chain. How high up does this center of mass move up? What is the change in energy?

If you marked off the center of mass (before pulling it up) and then pulled the chain so that the mark was on the table, the new center of mass would not be on the table. There would still be part of the chain hanging down which would have a center of mass below the table.

Forgive me for not being very concise but hopefully it helps.

5. Aug 14, 2014

Yashbhatt

But that's like sum of infinite series. You keep on finding the new center of mass and pulling it up.

6. Aug 14, 2014

Satvik Pandey

We can assume that total mass of chain hanging down the table is located at the CM of the part of chain below the table.You just need to find the work done in lifting that point(CM).

7. Aug 14, 2014

Yashbhatt

But even after the CM is on the table, you have to pull the remaining part.

8. Aug 14, 2014

Nathanael

There's always a complicated way to solve a problem.

But very often, as in this case, there is a simple way.

No energy is lost to friction, so conservation of energy will be useful to find the work done.

9. Aug 15, 2014

Yashbhatt

Okay. But I don't get why do have to assume that pulling only the center of mass is enough?

10. Aug 15, 2014

olivermsun

You don't assume that. What you do is compare the before-and-after potential energies of the system. You had to perform work on the system equal to the gain in potential energy, which is exactly determined by how much you've raised the center of the mass of the system.

11. Aug 15, 2014

ehild

See the picture. If the hanging part has n units (links) and each link has mass m and length ΔL you have to exert F=mg force to balance the weight of the hanging chain and do W(n)=(nm)g ΔL work to lift the chain by one link. The hanging part becomes one link shorter. To lift the next link, you exert (n-1)mg force and do W(n-1) = mg(n-1)ΔL work, and so on.

Initially, the hanging part consisted of N links. So your work to lift all the links of the chain is

W= W(N)+W(N-1).W(N-2)+.... +W(2)+W(1)= mgΔL[N+(N-1)+(M-2)+...2+1].

It is an arithmetic series. Yo know that 1+2+...(N-1)+N= (1+N)N /2≈ N22 if N >>1.

But m=M/N and ΔL = L/N, so the total work is W=(M/N)g(L/N) N2/2 = MGL/2, as if you pulled up a point mass M by length L/2.

ehild

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12. Aug 22, 2014

Yashbhatt

Great explanation. Just a question. You mentioned that the sum is N2 /2 if N >> 1. But what it's only a few links?

13. Aug 22, 2014

olivermsun

If N = 3, then the sum 1 + 2 + ... + N = 1 + 2 + 3.

14. Aug 22, 2014

ehild

You have to pull the last link only from half of its length. So W=mgΔL[N+N-1+N-2+....3+2]+mgΔL/2.

(You can grab the last link at the middle:tongue:)

ehild

15. Aug 22, 2014

Yashbhatt

But then one cannot say that it is equivalent to N2 / 2.

16. Aug 22, 2014

olivermsun

It is approximately true for N » 1 (N "large"), which is what ehild said.

17. Aug 22, 2014

Yashbhatt

And what I say is that what if you have 3 or 4 links hanging there. In that case, it won't be N2 / 2 but N(N + 1)/2. So, we won't get the L/2 thing.

18. Aug 22, 2014

olivermsun

You have a chain of mass M composed of 3 links, each with length ΔL, hence the total length of the chain L = 3ΔL.

Link #1's center of gravity is in the middle of the link, at height -ΔL/2. Therefore it needs to be pulled up by ΔL/2 and its potential energy increases by (M/3)gΔL/2.

Link #2 needs to be pulled up the length of the previous link, ΔL, plus another ΔL/2. Its potential energy increases by (M/3)g * 3ΔL/2.

Link #3 needs to be pulled up 2ΔL for the previous links and another ΔL/2. Its potential energy increases by (M/3)g * 5ΔL/2.

Total ΔPE = (M/3)gΔL/2 + 3(M/3)gΔL/2 + 5(M/3)gΔL/2 = Mg(1 + 3 + 5)/3 *ΔL/2 = Mg * 3ΔL/2 = Mg L/2.

19. Aug 22, 2014

ehild

You are right. In fact, you have to pull up the upmost link only by ΔL/2, while the others move up by ΔL.
So you grab the upmost link. You need to move it up only by ΔL/2 then its CM will be in level of the platform, and you lay it down, and move horizontally till the top of the next link reaches the platform. Your work is 2mgΔL+mgΔL/2. Lifting the next link needs mgΔL+mgΔL/2 work, and at last, your work is mgΔL/2. The total work is mgΔL[(2+1+0)+3˙1/2]=4.5 mgΔL.
In general, it is $W=mgΔL(\frac{N(N-1)}{2}+N\frac{1}{2})=\frac{N^2}{2}$

ehild

20. Sep 1, 2014

Murtuza Tipu

For such type of sum
general formula can be found out
work done =MgL/2n^2

where M is total mass of the whole chain
g is gravitational constant
L is total length of chain
n is the length of hanging chain in terms of L
Suppose L is 100 and length of hanging part is 25 then length of hanging part in terms of L is L/4 hence in this example n =4.
This formula can be derived easily.