Dynamics of a double atwood machine

[Aurelius120]

Homework Statement

In a double Atwood machine(pulley suspended from another pulley), a mass, m1 is suspended from one side of the upper pulley(on the other side is the lower pulley) and masses m2 and m3 are suspended from either side of the lower pulley. 𝑚3>𝑚2>𝑚1
𝑇 is the tension in the lower strings
𝑇1 is the tension in the upper strings
𝑎2 is the acceleration of 𝑚1 and of the lower pulley system (as a whole)
𝑎1 is the acceleration of 𝑚2 and 𝑚3 in the lower pulley (taking only the lower system into consideration)
𝑔 is the acceleration due to gravity
I want to find the net acceleration of each mass and the time for m1 to strike the upper pulley given the length of the string from m1 to the upper pulley is 20cm

Relevant Equations

N/A

I think the equations should be

𝑇−𝑚2𝑔=𝑚2𝑎1
𝑚3𝑔−𝑇=𝑚3𝑎1
𝑇1−𝑚1𝑔=𝑚1𝑎2
2𝑇−𝑇1=(𝑚2+𝑚3)𝑎2

P.S.- My textbook assumes T1 = 2T.
I don't understand why that is so.

 

[Doc Al]

Are the pulleys assumed massless?

 

[Chestermiller]

The accelerations of m2 and m3 are not equal, either in magnitude or direction. They would be equal in magnitude only if the lower pulley were not accelerating.

 

[haruspex]

𝑎1 is the acceleration of 𝑚2 and 𝑚3 in the lower pulley (taking only the lower system into consideration)

It is not clear what you mean by that. Since you are taking the two accelerations to have the same magnitude, I presume you mean these as accelerations relative to the pulley. But the pulley is accelerating, so this is a non-inertial frame. Consequently the equations below need another term:

𝑇−𝑚2𝑔=𝑚2𝑎1
𝑚3𝑔−𝑇=𝑚3𝑎1

P.S.- My textbook assumes T1 = 2T.

Consider the balance of forces on the lower pulley and write out the F=ma equation for it.

 

[Delta2]

Homework Statement:: In a double Atwood machine(pulley suspended from another pulley), a mass, m1 is suspended from one side of the upper pulley(on the other side is the lower pulley) and masses m2 and m3 are suspended from either side of the lower pulley. 𝑚3>𝑚2>𝑚1
𝑇 is the tension in the lower strings
𝑇1 is the tension in the upper strings
𝑎2 is the acceleration of 𝑚1 and of the lower pulley system (as a whole)
𝑎1 is the acceleration of 𝑚2 and 𝑚3 in the lower pulley (taking only the lower system into consideration)
𝑔 is the acceleration due to gravity
I want to find the net acceleration of each mass and the time for m1 to strike the upper pulley given the length of the string from m1 to the upper pulley is 20cm
Relevant Equations:: N/A

P.S.- My textbook assumes T1 = 2T

This holds only if the lower pulley is massless. To prove it proceed as @haruspex suggest and don't forget that the pulley is massless when applying Newton's 2nd law for this pulley.

Also a pulley has to be massless in order for the tensions on each side of the pulley to be equal. Just saying cause you might have taken this for granted, that it holds for all pulleys.

 

[Aurelius120]

Are the pulleys assumed massless?

yes they are massless and frictionless

 

[Aurelius120]

So will T1 be equal to 2T or will there be some other equations for T1?
[Because I think the string will slack if T1=2T
As m1 moves up with some a2 but the lower pulley will be stationary for T1=2T]

 

[Doc Al]

yes they are massless and frictionless

Good. From that you can deduce (not merely assume):

P.S.- My textbook assumes T1 = 2T.

 

[Doc Al]

So will T1 be equal to 2T or will there be some other equations for T1?

There may well be other equations in which T1 appears, but T1 = 2T follows from the fact that the pulley is massless.

[Because I think the string will slack if T1=2T
As m1 moves up with some a2 but the lower pulley will be stationary for T1=2T]

Not so.

 

[Doc Al]

(Apply Newton's 2nd law to the pulley itself. Since it's massless, the net force on it must be zero.)