Analyzing Forces and Acceleration in a System of Connected Particles

[mcintyre_ie]

Hey,
I’d appreciate some help with this question. Here’s a diagram:

Diagram

And the question:

A moveable pulley of mass 2Kg is suspended on a light inextensible string between two fixed pulleys as shown and masses of 6kg and 3kg are asttached to the ends of the string. If the system is released from rest:
(I) Show in a diagram the forces acting on each of the masses.
(II) Find the acceleration of the moveable pulley and the tension in the string.
(III) If initially the moveable pulley had been replaced by another of mass m, find m, given that the moveable pulley remains at rest while the other two masses are in motion

I’m a bit confused about the accelerations of each particle and the pulley. From what I understand, the 6kg particle will move downwards, and the pulley and 3kg particle will move upwards. I’m assuming that some particle (probably the pulley?) will have a different acceleration to the others, maybe twice that of the others? I’m just lost as to what the accelerations are.

(I)Should be fine if I can find the accelerations.

(II)Should be ok too, once I have the accelerations and then use F=MA.

(III)Seems complicated, and I’m feeling I’ll probably have difficulty.

So, any advice on the accelerations and part (III) would be appreciated.

Thanks in advance.

 

[Doc Al]

First label all the forces: the tension in the string plus the weights of the masses.
For simplicity, I would assume that the masses at each end accelerate downward: a₁ and a₂. (Note: Make an educated guess as to how it will move--if wrong, the acceleration will be negative.) The acceleration of the middle pulley is related to that of the masses by: a = (a₁ + a₂)/2 upward. Make sure you understand this. (Think what happens if both masses move down by a given distance--what happens to the middle pulley?)

 

[mcintyre_ie]

First label all the forces: the tension in the string plus the weights of the masses.
For simplicity, I would assume that the masses at each end accelerate downward: a₁ and a₂. (Note: Make an educated guess as to how it will move--if wrong, the acceleration will be negative.) The acceleration of the middle pulley is related to that of the masses by: a = (a₁ + a₂)/2 upward. Make sure you understand this. (Think what happens if both masses move down by a given distance--what happens to the middle pulley?)

I was thinking that the 6kg mass would be the only mass to move downwards, but both the other mass and the pulley would move upwards? For some reason I was thinking that the pulley would have an acceleration of half of that of each of the masses.

 

[Doc Al]

I was thinking that the 6kg mass would be the only mass to move downwards, but both the other mass and the pulley would move upwards?

No need to guess. Figure it out! To start, assume some direction for the acceleration of the the two masses. This will define your sign convention for applying F=ma. If you chose correctly, the accelerations will come out positive. Try it and see.

For some reason I was thinking that the pulley would have an acceleration of half of that of each of the masses.

I'm not sure what you mean. The motion of the pulley is related to the motion of the masses. See my previous post. To see the relationship, use some examples: if the left mass drops by a distance X₁, and the right one rises by X₂, then how must the pulley move? Don't forget that the rope is doubled over the middle pulley. (Note that I used your assumption of how the masses move.)