A pulley is hung from the ceiling on a rope. A block of mass M is suspended by a

Question

A pulley is hung from the ceiling on a rope. A block of mass M is suspended by another rope that passes over the pulley and is attached to the wall. The rope fastened to the wall makes a right angle with the wall. Ignore the masses of the rope and the pulley. Find (a) the tension in the rope from which the pulley hangs and (b) the angle that the rope makes with the ceiling.

Explanation / Answer

The tension on the pulley rope is Mgv2 The angle is 45° The tension in the main rope is Mg. The pulley is not accelerating, hence F_net=0. This means all vertical forces cancel and all horizontal forces cancel. If we call the tension in the pulley rope P: Vertical components: P·sin(?) - Mg = 0 Horizontal components: P·cos(?) - Mg = 0 -->Magnitude of P: P = v[(P·sin(?))² + (P·cos(?))²] = v[(Mg)² + (Mg)²] = Mgv2 -->Angle ?: sin(?) = Mg/P = Mg/(Mgv2) = 1/v2 ? = arcsin(1/v2) = 45°

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