Solving Massless Pulley: Finding Accelerations, Tension & Pulley Motion

[tonicandgin]

Homework Statement

m₁ and m₂ are connected by a massless string wrapped around a massless pulley. An external force F is applied to the pulley. m₁ does not equal m₂

find the acceleration of each mass, the tension in the string, and the acceleration of the pulley.

F external and m₁ and m₂ are known.

there is no gravity or friction in the problem.

Homework Equations

F=ma

The Attempt at a Solution

What I've worked out so far is that we must consider two reference frames to determine the accelerations. If we can calculate the acceleration of the pulley and the acceleration of one of the masses, the acceleration of the other mass should be determined.

If we consider the pulley's reference frame, one of the masses will be accelerating toward the pulley and the other mass will be accelerating with an equal and opposite acceleration.

If we consider m₁, we see the pulley accelerating toward m₁ and m₂ accelerating with an equal acceleration.

how can we combine these two reference frames to determine the accelerations in the lab frame?

is the acceleration of the center of mass of the system relevant?

 

[tiny-tim]

welcome to pf!

hi tonicandgin! welcome to pf!

… we must consider two reference frames …

no

call the position of the pulley x, call the length of string between the pulley and m₁ and m₂, respectively, L₁ and L₂ …

and use the fact that L₁ + L₂ must be constant …

what do you get? ​

 

[tonicandgin]

thanks for the response.

to clarify: what do you mean by "call the position of the pulley x"? is that its position before the force is applied and everything starts moving? or does x change as the system moves?


what i have so far is:

call position of the pulley x. call the length between pulley and the 2 masses L₁ and L₂, respectively
then position of m₁ = x - L₁ + 1/2 a_m1t²
and position of m₂ = x - L₂ + 1/2 a_m2t²

i've done a bunch of algebra trying to make use of L₁ + L₂ = L but nothing seems useful.

I think what i should do is write position equations for each mass and then take two time derivatives to get acceleration, but the only position equations i can write include the accelerations I'm looking for in the first place.

 

[tiny-tim]

hi tonicandgin!

(just got up :zzz: …)

… does x change as the system moves?

yes, that's the easiest way …

then you get …

position of m₁ = x - L₁
position of m₂ = x - L₂

then call the tension T, and do F = ma for each mass separately …

what do you get? ​

 

[tonicandgin]

thanks for the help. so far I have:

a₁ - a_r = a_p
a₂ + a_r = a_p

where a_r is the relative acceleration of each mass with the pulley.

( m₁x₁ + m₂x₂ ) / ( m₁ + m₂ ) is the center of mass of the system. if we take two time derivatives we get ( m₁a₁ + m₂a₂ ) / ( m₁ + m₂ ) which must equal the only external force, F.

F must = 2T

this is as much as i could get out of my professor today, he seemed to think it was solvable from here, but i still feel like there are too many unknowns and not enough equations.

 

[tiny-tim]

hi tonicandgin!

(just got up :zzz: …)

yes, there are two ways doing it

your can either start with F = ma for the centre of mass, as your professor suggests,

or you can just do F = ma for each of the three bodies separately

… i still feel like there are too many unknowns and not enough equations.

whichever method you choose, write out all the equations you have, and we'll see