Unit $5$ Progress Check: FRQ $1$

Show all your work for each part of the question. The parts within the question may not have equal weight.

A pulley is made from a solid disk that can rotate with negligible friction about a fixed axle at its center, as shown in Figure $1.$ The pulley has mass $M$ and radius $R,$ and its mass is distributed such that its rotational inertia is $\frac{1}{3}M{R}^2$ about its center. A string is wrapped around the pulley, and a downward force of magnitude $F_S$ is exerted on the end of the string.

$($a$)$

i$.$ The circle in Figure $2$ represents the pulley. On Figure $2,$ draw and label arrows that represent the forces $($not components$)$ that are exerted on the pulley while the force of magnitude $F_S$ is exerted on the string. Each force must be represented by a distinct arrow starting on$,$ and pointing away from, the point at which the force is exerted on the pulley.

Figure $2$

ii$.$ Using Newton's second law in rotational form, derive an expression for the tangential acceleration of a point located on the outer edge of the pulley. Express your answer in terms of $M,R\frac{,}{b}ar(F_S),$ and physical constants, as appropriate. Begin your derivation by writing either a fundamental physics principle or an equation from the reference booklet