# Homework Help: Two masses and two pulleys problem

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1. Sep 9, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Masses M1 and M2 are connected to a system of strings and pulleys as shown. The strings are massless and inextensible, and the pulleys are massless and frictionless. Find the acceleration of M1.

2. Relevant equations
Newton's 2nd Law of motion

3. The attempt at a solution

So here is my line of reasoning. We have four objects with which we can use Newton's second law to derive relationships which will get us an explicit expression of the acceleration of block 1.

Newton's law for block 1:
$T_1 - m_1 g = m_1 a_1$

Newton's law for block 2:
$T_2 - m_2 g = m_2 a_2$

Newton's law for pulley with block 2:
$T_1 - 2T_2 = 0$
$T_1 = 2T_2$

Next, if we combine the equation for block 1 with the equation for block 2, and if we assume $a_1 = a_2$, then we get,

$\displaystyle a_1 = \frac{g(2m_2 - m_1)}{m_1 - 2m_2}$

However, this is not the correct expression, because we must have that if M1 = M2 then $\displaystyle a_1 = \frac{g}{5}$

Where am I going wrong with my reasoning?

2. Sep 10, 2016

### ehild

a1 is not equal to a2. Think: the moving pulley sinks by dx, how much does m2 move?

3. Sep 10, 2016

### Mr Davis 97

But how does the second pulley move if it is massless?

4. Sep 10, 2016

### ehild

How does m2 move if the pulley does not move? The other end of the string is fixed to the ground.

5. Sep 10, 2016

### Mr Davis 97

Okay... But my equation $T_1 - 2T_2 = 0$ is correct since the mass of that pulley is zero?

6. Sep 10, 2016

### ehild

Yes.

7. Sep 10, 2016

### Mr Davis 97

Say for example that M1 is super heavy, and goes down. How does the second pulley accelerate upwards if it is massless?

8. Sep 10, 2016

### ehild

Why not? It pulls m2 upward.

9. Sep 10, 2016

### Mr Davis 97

But it's massless, so the net force on it is zero, which means there would be no acceleration it seems

10. Sep 10, 2016

### ehild

A massless object can accelerate without any force. F=ma. What can be a if both F and m are zero?

11. Sep 10, 2016

### Mr Davis 97

The tension on string 1 is twice the tension on string 2, so would it be fair to suppose that the acceleration of mass 2 would be twice that of mass 1?

12. Sep 10, 2016

### ehild

You get the relation between the accelerations from the constraints that the strings do not change their lengths.

13. Sep 10, 2016

### ehild

14. Sep 10, 2016

### Mr Davis 97

I'm just not seeing how to get the relation out of that... I know that I can say that the length of the string = so and so, and then differentiate twice... But I can't figure out the so and so.

15. Sep 10, 2016

### ehild

See the figure. If the pulley sinks by dx, the left piece of string becomes shorter by dx and the right one must become longer by dx. So the mass m2 moves downward by dx with respect to the pulley. But the pulley has moved downward by dx with respect to the ground, so the mass had to move by 2 dx downward with respect to the ground.

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16. Sep 10, 2016

### Mr Davis 97

How would you derive this from the fact that the length of the string is constant? If we say that $l = x_L + x_R + \pi r$, where r is the radius of the pulley, and the two x's are the left and right sections of the string, then if we differentiate twice we just get that $\ddot{x}_L = - \ddot{x}_R$, which doesn't show that the rate at which the mass moves is twice the rate at which the pulley moves.

17. Sep 10, 2016

### ehild

The acceleration is with respect to the ground. You need to differentiate h twice, instead of xr.

18. Sep 10, 2016

### Mr Davis 97

So is it something like $l = h + \pi r + (h - d_g)$ where $d_g$ is the distance of the block from the ground. So if we differentiate twice we get that $\ddot{d_g} = 2\ddot{h}$, which means that the rate at which the block accelerates is twice the rate at which the pulley accelerates...?

19. Sep 10, 2016

### ehild

Yes, and twice the acceleration of the other mass, only opposite direction.

20. Sep 10, 2016

### Mr Davis 97

Do you get this from looking at the constant length of the other string for the other pulley?

21. Sep 10, 2016

### ehild

Yes. Because of the length of the other string is also constant, and the left pulley is fixed, the accelerations of the hanging pulley and mass m1 are opposite and of equal magnitude.

22. Sep 10, 2016

### Mr Davis 97

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Last edited: Sep 10, 2016
23. Sep 10, 2016

### Mr Davis 97

I have one more question about the nature of massless pulleys. If they are massless, wouldn't their acceleration be infinite according to the equation $a = \frac{F}{m}$? How can we justify letting the masses of pulleys and ropes be zero?

24. Sep 10, 2016

### ehild

That is why the net force on a massless pulley has to be zero. Otherwise the acceleration would be infinite. But we know that the pulley has finite acceleration. Therefore the net force is zero.

25. Sep 10, 2016

### Mr Davis 97

But if the pulley has a finite acceleration, how would the net force be zero?

Last edited: Sep 10, 2016