Problem $4(10$ pts$)$:

The system shown in the figure below is comprised of a disk of radius $R,$ with mass $m,$ center point $O,$ and mass moment

of inertia about the center of mass $I_O$ that rolls without slipping, with point of contact at $Q.$ When the spring, attached a

distance $b$ from the center of the disk, is unstretched, $=0.$ An input torque is applied by a motor at the point $O.$ is

considered a small angle for this problem. Gravity acts downward.

Your tasks:

A$.$ Derive the equations of motion for the system in terms of $,0^{},$ etc. Remember all your steps $-$ free body diagrams, etc.

$(10$ pts$)$

B$.$ BONUSI If the input torque, $(t)$ is equal to at $\frac{N}{m},$ Is the small angle approximation actually valid for this problem?

Justify your answer for full credit. $(4$ points$).$

Can you solve this using the instantaneous center of velocity at point Q$.$ Thank you