Answer to Question #247622 in Physics for asyiqin

Question

A pulley is mounted on a frictionless axis and then the pulley is attached to the

higher end of the inclined surface as shown in the figure. A massless cord is

wrapped around the pulley while its other end is tied to a 1.0 kg object. The pulley

has a radius of 20 cm and moment of inertia of 0.4 kg m2

. The angle of the surface is 37° and the coefficient of kinetic friction is 0.2. The object is released from rest. The pulley starts to rotate as the object moves and the cord unwinds.

Determine the acceleration of the object and the tension in the cord.

Answer 1

Forces on the object:

$$ma=-\mu mg\cos\theta+mg\sin\theta-T.$$

Torque the tension creates:

$$I\alpha=Tr,\text{ where }\alpha=\frac ar.$$

Substitute:

$$I\frac ar=[mg(\sin\theta-\mu\cos\theta)-ma]r,\\\space\\ I\frac ar=mgr(\sin\theta-\mu\cos\theta)-mar,\\\space\\ a\bigg(\frac Ir+mr\bigg)=mgr(\sin\theta-\mu\cos\theta),\\\space\\ a=\frac{mgr^2(\sin\theta-\mu\cos\theta)}{I+mr^2}=0.39\text{ m/s}^2.\\\space\\ T=mg(\sin\theta-\mu\cos\theta)-ma=3.94\text{ N}.$$