Double Atwood Machine (With one end of the string connected to the ground)

[mikee]

1. Homework Statement [/b
]http://teacher.pas.rochester.edu/phy121/PracticeExams/FinalExam/FinalExam_files/image015.jpg
This is not the exact picture, instead of having M1 there let end of the string be connected to the ground instead
Find the tensions and accelerations

Homework Equations

The Attempt at a Solution

Hello, i am having a little problem with this question, i attempted to let T3=2(T1), because i thought the normal force tension from the ground and tension from mass 2 would give a tension equal to T1, and then i looked at it again and thought to myself that the acceleration of mass 2 would have to be the same as the acceleration of mass 3 by the conservation of string, because since one end of the second pulley is connected to the ground, the ground will never move up or down so therefore only mass 2 will be able to move and it will have to move the same length as mass 3 and therefore the accelerations will be the same, but that doesn't add up if the tensions are different how can the accelerations be the same, could someone please tell me if i am missing somthing?

 

[tiny-tim]

Hello mikee!

i attempted to let T3=2(T1) …

ah, you're saying that the small pulley has only three forces acting on it, which are T₃ and T₁ twice, and so T₃ = 2T₁.

but that only works if the small pulley is not accelerating (and has mass negligible compared with the blocks)

 

[tomcue]

I would say that your thought process is on the right track. However, you are missing a key concept in this scenario - the mass of the pulley. In a double Atwood machine, the mass of the pulley can affect the tensions and accelerations of the masses. In this case, since one end of the string is connected to the ground, the pulley will not be able to move and therefore, its mass will not contribute to the overall system.

To find the tensions and accelerations, you can use the equations for a double Atwood machine:

T1 - T2 = m1a1
T3 - T4 = m2a2

Since T3 = 2T1 (as you correctly assumed), we can rewrite the equations as:

T1 - T2 = m1a1
2T1 - T4 = m2a2

Now, we also know that the acceleration of the pulley (a3) is equal to the acceleration of mass 1 and mass 2 (a1 and a2) because of the conservation of string. Therefore, we can add another equation to our system:

a1 = a2 = a3

Now, we have three equations and three unknowns (T1, T2, and T4). We can solve for these unknowns using algebraic manipulation. Once we have the tensions, we can use them to find the accelerations using the equations above.

I would also recommend drawing a free-body diagram for each mass and the pulley to help visualize the forces acting on each object. This can also help you in solving the equations.

I hope this helps and good luck on your homework!