# Frames of reference in a pulley system

This question comes from the following website: https://i-want-to-study-engineering.org/q/pulley_dynamics_1/

A system consists of three light pulleys and light inextensible strings as shown in the figure. If the upward accelerations of A and D are 4m/s^2 and −4m/s^2 respectively and B is not accelerating relative to the global coordinate system, calculate the upward acceleration of C.

The video explanation for the solution starts off by stating:

The left hand side of this system has two constraints. Mass A has an acceleration of 4m/s^2 and mass B has zero acceleration. Looking at Point P, these constraints mean that for every 2 units that mass A is elevated by, the point P increases in height by 1 unit. Hence the absolute acceleration of the connecting string is 2m/s^2

I am afraid that I do not understand how the acceleration of masses A and B being 4 and 0 respectively implies that their combined absolute acceleration is 2.

Is the absolute acceleration of A and B not already 4 and 0? I am bamboozled.

• COMBINED absolute acceleration is 2, remember the pulley above them will move with half the acceleration of each. Half of 4 and 0 is 2 – Adrian Howard Oct 7 at 15:27
• The statement in bold is not true in general. The acceleration of P is half that of A, but since we aren’t told initial velocities, we can’t say that displacements follow that ratio. – Ben51 Oct 7 at 15:55

First examine left side of big central pulley. Point P must have some acceleration $$a$$ since A is accelerating and B is not. Since strings are not elastic and all masses are equal (I assume), it follows:

$$a_A + a = 4m/s^2$$,

$$a_B + a = 0m/s^2$$,

$$a_A = - a_B$$

which leads to $$a = 2m/s^2$$, $$a_A = 2m/s^2$$, $$a_B = -2m/s^2$$ ($$a_A$$ and $$a_B$$ are expressed in P frame of reference)

On the right side of big pulley:

$$a_D - a = -4$$ => $$a_D = -4+2 = -2m/s^2$$

and therefore:

$$a_C - a = -a_D - a = 2-2 = 0m/s^2$$

• okay this makes a lot of sense thank you! – David Toh Oct 8 at 1:27

Think in terms of distances moved.

If $$A$$ moves up 4 metres and $$B$$ does not move at all then 4 metres of “slack” string is produced.

The pulley and point $$P$$ has to move up 2 metres with 2 metres of slack string taken up on each side of the pulley to make the string taut.

The same ratios apply to the speeds and the accelerations of the masses and the pulley and point $$P$$.

• What do you mean by “each side of the pulley”? – David Toh Oct 7 at 16:21
• Say I only consider the pulley system that consists of masses A and B. Would mass A be accelerating upwards at 2m/s^2 and mass B be accelerating at -2m/s^2? – David Toh Oct 7 at 16:32
• But since point P is accelerating at 2m/s^2, the net absolute acceleration of A and B is 4 and 0? – David Toh Oct 7 at 16:35
• You are told that $B$ is not accelerating. – Farcher Oct 7 at 17:33