Question: Solving for Accelerations in an Atwood Machine System

[Asterast]

Summary:: I have solved the question but I'm getting answers wrong, some reaction equations seems to have trouble.

Question: Let m₁ = 1 kg, m₂ = 2 kg and m₃ = 3 kg in figure. Find the accelerations of m₁, m₂ and m₃. The string from the upper pulley to m₁ is 20 cm when the system is released from rest. How long will it take before m₁ strikes the pulley?

here's what I did:
Assuming acceleration of m₁ be a
Assuming acceleration of m₂ and m₃ be a'
Tension in string between mSUB]2[/SUB] and m₃ be T
so tension in upper string connected to m₁ is 2T
So equations I made for the bodies:

equation for m₁
2T-m₁g = m₁a

2T-g=a ...

equation 1

equation for m₂
T-m₂g = m₂(a'-a)
Since m₂ will be going up because of weight of m₃ and system of mm₂ and m₃ will be going downwards.

T-2g=2(a-a') ...

equation 2

equation for m₃
m₃g-T = m₃(a'+a)
since m₃ will be going down simultaneously with m₂ and m₃

3g-T=3(a'+a) ...

equation 3

_________
Eliminating T from equation 1 and 2

3g = -3a+4a' ...

equation 4

_________
Eliminating T from equation 2 and 3

g = 5a+a' ...

equation 5

Solving equation 4 and 5 I get a' = 18g/23
Which is wrong.. please help. :3

 

[BvU]

Summary:: I have solved the question but I'm getting answers wrong

Solved ?

Assuming acceleration of m2 and m3 be a'

Do you really think they undergo the same acceleration ?

 

[BvU]

so tension in upper string connected to m1 is 2T

Your conclusion must then be: lower pulley does not move

 

[Asterast]

Solved ?
Do you really think they undergo the same acceleration ?

I assigned a' for the lower string system since motion of the rate of motion of m2 will be equal to m3
their final accelerations will be a'-a and a'+a for m2 and m3 respectively.

 

[Asterast]

Your conclusion must then be: lower pulley does not move

How come? lower string has tension T and T on each side so upper string will have 2T.
Also check the equations of m2 and m3, their final accelerations are a-a' and a+a'

 

[BvU]

I assigned a' for the lower string system since motion of the rate of motion of m2 will be equal to m3
their final accelerations will be a'-a and a'+a for m2 and m3 respectively.

Sounds much better than

Assuming acceleration of m2 and m3 be a'

Your conclusion must then be: lower pulley does not move

My bad: it is massless.

Back to my drawing board

 

[BvU]

I assigned a' for the lower string system

And a' is different from a ? Still ambiguous

 

[BvU]

motion of the rate of motion of m2 will be equal to m3

With respect to what ?

 

[Asterast]

And a' is different from a ? Still ambiguous

Shouldn't it be different because m2 and m3 will be moving wrt to each other because m3>m2 with acceleration a'
The system of m3 and m2 will be moving wrt m1 with acceleration a.

 

[Asterast]

Shouldn't it be different because m2 and m3 will be moving wrt to each other because m3>m2 with acceleration a'
The system of m3 and m2 will be moving wrt m1 with acceleration a.

And their final accelerations are equated in eq1,eq2 and eq3

 

[ehild]

equation for m₁
2T-m₁g = m₁a

2T-g=a ...

equation for m₂
T-m₂g = m₂(a'-a)*****
Since m₂ will be going up because of weight of m₃ and system of mm₂ and m₃ will be going downwards.

T-2g=2(a-a') ...*****

The equations with stars contradict to each other,
The first one is correct. The right side of the upper string goes downward, m2 goes up with acceleration a' with respect to the pulley, so its upward acceleration is a'-a.
Check also the equation for m3.

 

[Asterast]

The equations with stars contradict to each other,
The first one is correct. The right side of the upper string goes downward, m2 goes up with acceleration a' with respect to the pulley, so its upward acceleration is a'-a.
Check also the equation for m3.

I'm sorry must have been typing mistake. Let me fix it and check the solution I've solved on paper real quick.

 

[Asterast]

The equations with stars contradict to each other,
The first one is correct. The right side of the upper string goes downward, m2 goes up with acceleration a' with respect to the pulley, so its upward acceleration is a'-a.
Check also the equation for m3.

Hello
So the first starred equation was a typo for me. Actually I didn't know that acceleration for m2 should be a'-a, I have been trying to solve the problem by taking the acceleration a-a'; because m2 will be going down even though it is moving upwards hence I took a-a' can you please explain me why it shouldn't be a-a'

 

[ehild]

Hello
So the first starred equation was a typo for me. Actually I didn't know that acceleration for m2 should be a'-a, I have been trying to solve the problem by taking the acceleration a-a'; because m2 will be going down even though it is moving upwards hence I took a-a' can you please explain me why it shouldn't be a-a'

Ser picture

It is sure that m1 accelerates upward, with a. The moving pulley accelerates downward with a. m3 accelerates downward with a' with respect to the moving pulley , as it is heavier than m2, and . m2 accelerates upward with a' with respect to the moving pulley. Taking upward the positive direction relative to the ground, acceleration of m2 is a'-a , and that of m3 is -(a'+a).
You can not know beforehand if m2 moves up or down.

 

[Asterast]

Ser picture
View attachment 258338
It is sure that m1 accelerates upward, with a. The moving pulley accelerates downward with a. m3 accelerates downward with a' with respect to the moving pulley , as it is heavier than m2, and . m2 accelerates upward with a' with respect to the moving pulley. Taking upward the positive direction relative to the ground, acceleration of m2 is a'-a , and that of m3 is -(a'+a).
You can not know beforehand if m2 moves up or down.

Isn't m2 actually going downwards overall, even though it's moving upwards wrt m3.
So direction of acceleration of m2 should be downwards??

 

[ehild]

Isn't m2 actually going downwards overall, even though it's moving upwards wrt m3.'
So direction of acceleration of m2 should be downwards??

You can not know. and it does not matter. The acceleration of m2 with respect to the ground is a'-a., it goes upward if a'>a and goes downward if a'<a. In this case you can say that m2 accelerates downward with a-a'. You can not decide with logic alone. Solve for a and a' and you will see.
You have to fit the sign of the forces to the direction of acceleration.

 

[Asterast]

You can not know. and it does not matter. The acceleration of m2 with respect to the ground is a'-a., it goes upward if a'>a and goes downward if a'<a. In this case you can say that m2 accelerates downward with a-a'. You can not decide with logic alone. Solve for a and a' and you will see.
You have to fit the sign of the forces to the direction of acceleration.

Thank you so much I got it.