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Lets look at the individual particles and the force diagrams for each

Note that the friction force F_r and the Tension in String 1 work together and opposite the the other Tension.

The key in this stage would be that the tension in both strings would be different. Usually we might represent them as T & S ... when we compile all the equations of Force we will clear all the Tensions and be left with a formula in a & in g.

Previous to this section we always intuitively knew that the particles will go in direction of the greatest force.

This still applies, but it is less intuitive, if it was just the blue 13 kg particle and the green particle it would fall down on the blue side.

The reason we can compare these particles like by like is because the other factors are the same, they have only one pulling force acting upon them.

Look at the red particle .... it has 2 pulling forces acting on it, and so in essence the mass is divided across the 2 forces. This leaves the effective weight per string at 10g .... which is greater than the 8g, but less than the 13g.

The direction of the acceleration between the blue particle and the red particle is down with the blue particle, and up with the red particle.

The red particle vs the green particle, the acceleration here drives the red particle downwards and the green particle goes up.

If the acceleration of the blue particle is given by a and the green particle acceleration given by f, then the acceleration of the red particle is given by (a + f)/2

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