Struggling to Understand Pulley System Dynamics

[JoshW]

Homework Statement

it is Problem 3.30.

Homework Equations

I am not sure how to sum the forces about each consecutive pulley I have been at this for 4 hours now. It is sad I can solve the DE questions and not this.

The Attempt at a Solution

I had the first pulley at the top as (where y is dv/dt)
-T+mg=my₁

Then I figured if i did it with just one extra pulley I could model the rest how ever it is not correct.

-2T+mg=-my₂ (as i set the up direction to be negative and this mass would go up)

Then with two pulleys with the string under it I had either

-3T+mg=-ma or -4T+mg=-my₂

What I don't get is how the tension from each consecutive pulley connected to the center of the one above it will effect the system.

 

[JoshW]

 

[haruspex]

The curve of the page makes the end of one line (the number of pulleys) illegible.

Just consider one of the pulleys with the string passing beneath it. If the tension in the string going up from its right is T_n, what is the tension in the string going up from its left hand side? What is the tension in the string going down from its centre?

 

[JoshW]

it says consider N pulleys. so the tension of the one come out of the center would be the sum of the two above it? then string to the left would be just T?

 

[haruspex]

it says consider N pulleys. so the tension of the one come out of the center would be the sum of the two above it? then string to the left would be just T?

Right. So let the tension in the string passing under the n^th pulley be T_n. What is the relationship between T_n and T_n+1?