# More complicated systems of pulleys

A more complicated system is shown in Figure 1. Here the free central pulley is thought of as having a mass; once again, the string is light, and all pulleys smooth. Figure 1: system of three pulleys, two fixed and one free.

Suppose, for example, that the mass A is 1 kg, the mass B is 3 kg and the mass of the free pulley is 2 kg. Suppose that A accelerates upwards with acceleration a_{1} (if the acceleration is downwards, this will come out negative) and the B accelerates upwards with acceleration a_{2}.

Then a key observation is that the downward acceleration of the free pulley is \left(a_{1}+a_{2}\right)/2. This isn't obvious, so let's think about why it's the case.

The reason is that if the section of the string from which A is suspended shortens by x_{1} metres, and that from which B is suspended by x_{2} metres, then the central loop lengthens by \left(x_{1}+x_{2}\right) metres.

Since this loop consists of two vertical sections (and a semicircular one around the pulley whose length stays constant), this means that the pulley descends a distance of \left(x_{1}+x_{2}\right)/2 metres.

Simply differentiating twice gives us the result that the downward acceleration of the pulley is

\frac{d^{2}}{d\,t^{2}}\left(\frac{x_{1}+x_{2}}{2}\right)=\frac{a_{1}+a_{2}}{2}.

Figure 2 shows the system with the forces and accelerations added. Figure 2: System of pulleys showing forces and accelerations

Balancing forces about the mass A:

T-g=a_{1}.

Balancing forces about the mass B:

T-3g=3a_{2}.

Balancing forces about the free pulley:

2g-2T=2\frac{a_{1}+a_{2}}{2}=a_{1}+a_{2}.

Subtracting the first equation from the third gives:

3g-3T=a_{2}.

Adding this equation to three times the second gives

-6g=10a_{2}

This gives us a_{2}=-3g/5. From the second equation we get that

T=-\frac{9g}{5}+3g=\frac{6g}{5}.

Now, from the first equation,

a_{1}=\frac{6g}{5}-g=\frac{g}{5}.

Finally, the downward acceleration of the free pulley is

\frac{a_{1}+a_{2}}{2}=-\frac{g}{5}.

So all in all, the mass A accelerates upwards with acceleration g/5, the mass B accelerates downwards with acceleration 3g/5 and the free pulley accelerates upwards with acceleration g/5. This behaviour is shown in the animation. Animation 1: Motion of a system of pulleys

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