Pulleys  (Page 4/5)

Now, the lengths of the two strings are constant. Let they be $L_1\text{and}{L}_2$ .

$$\begin{array}{l}{L}_1=x_0+x_1\\ L_2=(x_2-x_0)+(x_3-x_0)=x_2+x_3-2x_0\end{array}$$

Eliminating $x_0$ , we have :

$$\begin{array}{l}\Rightarrow L_2=x_2+x_3-2L_1+2x_1\\ \Rightarrow x_2+x_3+2x_1=2L_1+L_2=\text{constant}\end{array}$$

This is the needed relation for the positions. We know that acceleration is second derivative of position with respect to time. Hence,

This gives the relation of accelerations involved in the pulley system.

2: Tensions in the strings

We can, now, find the relation between tensions in two strings by considering the free body diagram of pulley “B” as shown in the figure.

Here,

3: Free body diagrams of the blocks

The free body diagrams of the blocks are as shown in the figure.

Thus, we have altogether 5 equations for 5 unknowns.

There is one important aspect of the motions of blocks of mass " $m_2$ " and " $m_3$ " with respect to moving pulley "B". The motion of blocks take place with respect to an accelerating pulley. Thus, interpretation of the acceleration must be specific about the reference (ground or moving pulley). We should ensure that all measurements are in the same frame. In the methods, described above we have considered accelerations with respect to ground. Thus, if acceleration is given with respect to the moving pulley in an analysis, then we must first change value with respect to the ground.