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?

The test for motion compares the net driving force with the limiting friction. Down the plane A is pulled by $6mg\sin30^{\circ}=3mg$ and held back by the tension, which cannot exceed B's weight $2mg$ while B is on the point of rising, leaving $mg$ to overcome friction. The limiting friction uses A's own normal reaction, $R=6mg\cos30^{\circ}$, so it is $0.25\times6mg\cos30^{\circ}=1.299mg$. Since $mg<1.299mg$, the system stays at rest. Comparing $3mg$ with $2mg$ ignores friction altogether, and the friction depends on the mass on the plane, not on the hanging mass.