Forces- two pulleys w. diff accelerations

[Maiia]

Homework Statement

In the drawing the rope and the pulleys are massless and frictionless, and there is no friction between the table and the block. Find the tension in the rope and the acceleration of the 10kg block.

Here is my drawing of the problem and my FBDs and equations.

I assumed that if the acceleration of the 10kg block was "L," the acceleration of the 3kg block would be 1/2 L because it is split between the two tensions...?

I would appreciate it if someone could check my work for this problem because I don't seem to be getting the right answer...

 

[alphysicist]

Hi Maiia,

Homework Statement

In the drawing the rope and the pulleys are massless and frictionless, and there is no friction between the table and the block. Find the tension in the rope and the acceleration of the 10kg block.

Here is my drawing of the problem and my FBDs and equations.


I assumed that if the acceleration of the 10kg block was "L," the acceleration of the 3kg block would be 1/2 L because it is split between the two tensions...?

I would appreciate it if someone could check my work for this problem because I don't seem to be getting the right answer...

In your equation for the 3kg mass, you have:

$$2T - F_g = \frac{3}{2} L$$
where L is the acceleration of the 10kg block. You seem to have a sign error here. Remember that everything with the same sign is going in the same direction, which would mean that this equation is saying the acceleration of the 3kg mass is in the same direction as the tension (and in the opposite direction of the force of gravity).

 

[thomate1]

Thank you for sharing your work and thought process on this problem. It is always helpful to see how others approach and solve physics problems.

From my understanding of the problem, it seems like you have correctly identified the forces acting on the system and set up the appropriate free body diagrams. However, I would like to clarify a few points to ensure accuracy in your solution.

Firstly, you are correct in assuming that the acceleration of the 3kg block would be half of the 10kg block's acceleration, as the tension in the rope is split between the two blocks. However, this does not necessarily mean that the acceleration of the 3kg block is 1/2 L. It would depend on the values of the tensions acting on the block.

Secondly, it is important to note that the tension in the rope is not the same throughout its length. In this case, the tension is different on either side of the pulleys due to the difference in the accelerations of the blocks. This is known as the "tension gradient" in the rope.

To accurately solve for the tension and acceleration of the 10kg block, you would need to set up and solve a system of equations using the Newton's second law for each block, taking into account the tension gradient in the rope. This would give you two equations with two unknowns (tension and acceleration). Solving these equations would give you the correct values.

I hope this helps in your understanding of the problem. Keep up the good work!