Connected Particles Questions and Answers

A particle A of mass $6m$ kg rests on a rough plane inclined at $30^{\circ}$ to the horizontal, with coefficient of friction 0.25. A light inextensible string runs from A over a smooth pulley at the top of the plane to a particle B of mass $2m$ kg hanging freely. The system is released from rest. Does A slide down the plane?

A light inextensible string carrying tension T passes over a smooth pulley P fixed at the top of a plane inclined at $30^{\circ}$ to the horizontal. One part of the string runs down the slope to a particle on the plane, and the other hangs vertically. What is the magnitude of the force exerted by the string on P?

A toy car P (mass 4 kg) rests on a rough horizontal plane, with coefficient of friction $\frac12$ between P and the plane, and is connected by a light inextensible string over a smooth pulley X to toy car Q (mass 8 kg) held on a smooth plane inclined at 30° to the horizontal. When Q is released, find the acceleration of the system and the tension in the string, in terms of $g$.