Answer to Question #245873 in Physics for Eyo

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

The forces the sliding body creates:

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

The toques the forces create on the pulley:

$$I\alpha=Tr.$$

We remember that

$$a=\alpha r:\\\space\\ I\frac ar=[mg(\sin\theta-\mu\cos\theta)-ma]r,\\\space\\ I\frac ar=mgr(\sin\theta-\mu\cos\theta)-mar,\\\space\\ a\bigg(\frac Ir-mr\bigg)=mgr(\sin\theta-\mu\cos\theta),\\\space\\ a=\frac{mgr^2(\sin\theta-\mu\cos\theta)}{I-mr^2}=0.48\text{ m/s}^2.\\\space\\ T=mg(\sin\theta-\mu\cos\theta)-ma=3.85\text{ N}.$$