Derive the displacements movable pulleys.

[Legendon]

Homework Statement

Obtain the accelerations of mass a and mass b.

Homework Equations

Once i derive the displacement relation constraint, then i just have to diff twice w.r.t time. This part is ok.

The Attempt at a Solution

I am trying to reason this way: suppose 2M moves up by x. Following the right string of 2M to pulley2 then to pulley 1 = This means pulley1 move down by x. Following the left string of 2M to pulley1 = this causes M to move down by x. So M moves by 2x. the acc of m must be 2a and acc of 2m is a. but the answer is m acc at 3a and 2m acc at a.

 

[ehild]

The right piece of the left string gets shorter by 2x when the block B rises by x, as the left pulley descends by x. So the left block moves down by 2x with respect to the left pulley.

ehild

 

[Legendon]

Ok, I'm getting it. The right piece of left string gets shorter by 2x meaning that the left piece of the right string must increase by 2x thus left block moves 2x w.r.t left pulley. Since left pulley is moving at dist x w.r.t right pulley. left block is moving 3x wrt right pulley. this explains the 3:1 acc ratio. Am I right.
Thanks

 

[ehild]

It is right now.

ehild

 

[rwisz]

I would first clarify the problem and make sure I understand the given information correctly. It seems that the problem involves two masses (M and 2M) connected by a system of movable pulleys. The goal is to find the accelerations of each mass.

To solve this problem, I would use the fundamental principles of mechanics, such as Newton's laws of motion and the equations of motion. I would also draw a free body diagram for each mass and apply the appropriate equations to determine the accelerations.

In this case, since the system involves movable pulleys, I would also need to consider the constraint equations that relate the displacements of the masses and the pulleys. These constraints would help me determine the relationship between the displacements and accelerations of the masses.

It seems that the student has attempted to use the displacement relation constraint to determine the accelerations of the masses. However, their reasoning may not be entirely correct, as the displacement relation constraint does not directly relate to the accelerations of the masses. Instead, it relates the displacements of the masses to the displacements of the pulleys.

To correctly solve this problem, I would recommend deriving the equations of motion for each mass, taking into account the constraint equations and the forces acting on each mass. This would result in a system of equations that can be solved to determine the accelerations of the masses.

In summary, as a scientist, I would approach this problem by carefully analyzing the given information, applying the relevant principles and equations, and using logical reasoning to arrive at the correct solution.