# Work Question (Pulling a chain on the Moon)

1. Feb 1, 2017

### Arman777

1. The problem statement, all variables and given/known data
At a lunar base,a uniform chain hangs over the edge of a horizontal platform.A machine does $1.0J$ of work in pulling the rest of the chain onto the platform.The chain has a mass of $2.0 kg$ and a length of $3.0m$.What lenght was initally hanging over the edge ? On the moon,the gravitational acceleraiton is $\frac 1 6$ of $9,8 \frac {m} {s^2}$
2. Relevant equations
$ρ_{chain}=M/L$
$W_g=-ΔU_g$

3. The attempt at a solution
Lets suppose $L_1$ lenght is needed to pull.
So chain density will be $ρ=\frac {2kg} {3m}=0.66\frac {kg} {m}$
The mass of $L_1$ is$L_1ρ=M_1$

The gravitational work on $M_1$ is $W_g$ which its $-M_1gL_1=1J$
$(L_!)^2ρg=6J$

which $L_1=1.05m$ but the answer is $1.4m$

Theres a "-" sign which makes me uncomfortable.Also I am making wrong at some point

2. Feb 1, 2017

### Staff: Mentor

Hint: If the length hanging over the edge is L, does every bit of the mass have to pulled up a height L?

Also, the work is done by some force pulling the chain up, so its sign would be +. (It's work against gravity, not by gravity.)

3. Feb 1, 2017

### Arman777

Here what I did

$(M_1dm)g(L1-dl)=W-dw$ dw is zero and $(M_1-dm)=(L_1-dl)p$
Is this true ?
yeah thats right but the work done by gravity on the chain is negative .The work done by machine to the chain is positive.Thats why I confused

4. Feb 2, 2017

### Staff: Mentor

Not quite sure what you're doing here. Are you trying to set up an integral?

Hint: If some extended object changed height, what point on the object would you track to compute its change in gravitational PE?

The work done by the machine equals the change in gravitational PE; both are positive.

5. Feb 2, 2017

### Arman777

yep I see now
I was...If the lengh decreases a bit what would be happen
Ok let me try again

6. Feb 2, 2017

### Arman777

I found 1.36m by usig this equation
$(L_1)^2ρg=12J$
Is it true ?

7. Feb 2, 2017

### Staff: Mentor

Don't just toss out an equation; show how you got the equation.

Consider a mass element dm of the hanging section of the chain. If the length hanging is L, what is the average distance that each mass element must be lifted to get to the platform?

8. Feb 2, 2017

### Arman777

$M_1a_gH=1J$

$H=\frac {L_1} {2}$; The center of mass of $M_1$ moves this much.
$M_1=L_1ρ$
so,
$(L_1)ρ\frac {g} {6}\frac {L_1} {2}=1J$

9. Feb 2, 2017

### Staff: Mentor

Perfect!

10. Feb 2, 2017

### Arman777

Average distance will be $\frac {L_1} {2}$

$∫\frac {L_1} {2}dmg=1J$ from 0 to M and so This is true I am sure but If I wanted to convert it to $dl$ ,

$dm=dlρ$ so ;
$∫\frac {L_1} {2}gρdl$ from 0 to $L_1p$ ?

Last edited: Feb 2, 2017
11. Feb 2, 2017

### Staff: Mentor

Once you use the center of mass there's no need to integrate. But if you do, you'll get the same answer.

12. Feb 2, 2017

### Arman777

Oh ok thank you