Two Blocks and Two Pulleys

Learning Goal:

To be able to calculate the tension in a string and the acceleration of each of two blocks in a two-pulley system.

As shown, a block with mass m1 is attached to a massless ideal string. The string wraps around a massless pulley and then wraps around a second massless pulley that is attached to a block with mass m2 and ultimately attaches to a wall. The whole system is frictionless.

A coordinate system is given in the figure. (Figure 1) 

 Tension in the string

Given that a2 is the magnitude of the horizontal acceleration of the block with mass m2, what is T, the tension in the string?

Express the tension in terms of m2 and a2.

T =

m2a22

Acceleration of suspended block

Given T, the tension in the string, calculate a1, the magnitude of the vertical acceleration of the block with mass m1.

Express the acceleration's magnitude, a1, in terms of m1, g, and T.

a1 =	∣∣T−m1gm1∣∣

Relative acceleration between blocks

Given the magnitude a1 of the acceleration of the block with mass m1, find a2, the magnitude of the horizontal acceleration of the block with mass m2.

Express a2 in terms of a1.

 

a2 =

a12

 

Using the result from Part C in the equation for T from Part A, express T as a function of a1.

Express your answer in terms of m2 and a1.

T =

m2a14

Having solved the previous parts, you have all the pieces needed to calculate a1, the magnitude of the acceleration of the block with mass m1. Enter an expression for a1.

Express the acceleration's magnitude a1 in terms of m1, m2, and g.

a1 =	4m1gm2+4m1