Answer to Question #246539 in Physics for Hyee

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

Answer 1

According to Newton's second law for the block:

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

By Newton's second law for rotational motion:

$$I\alpha=Tr.$$

Write the equation that relates the angular and tangential acceleration, substitute, and solve for tension:

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