A spool with mass $m$ and outer radius R rolls without slipping on a horizontal surface. A block of

mass $M$ is in contact with the inner radius, $r,$ of the spool and is constrained to move on a

smooth, horizontal surface. The block does not slip on the spool. A spring with stiffness $k$

connects the center of the spool to ground. The coordinate $x,$ where $x$ is measured as shown, is

zero when the spring is unstretched. $($Note that this is a single degree of freedom system.$)$

a$.$ Draw the free body diagrams of both the spool and the block.

b$.$ Use the Newton$-$Euler approach to determine the equation of motion for the system.

HINT: Use the sum of the moments about point G to help eliminate the friction terms

from the equation of motion.

c$.$ Use the Power Equation to formulate the equation of motion. HINT: Remember that

the friction forces don't do work on the system!

d$.$ Are your answers to part b and part c identical? HINT: Your answer should be yes!