$?($a$)$ A any string, how fast would its center be

moving when it had fallen the distance in

part $($a$)?($c$)$ Why do you get two different

answers when the cylinder falls the same

distance in both cases?

$10\mathrm{.}30$ A Ball Rolling Uphill. A bowling ball rolls without slip$-$

ping up a ramp that slopes upward at an angle $$ to the horizontal $($see

Example $10\mathrm{.}7$ in Section $10\mathrm{.}3).$ Treat the ball as a uniform solid sphere,

ignoring the finger holes. $($a$)$ Draw the free$-$body diagram for the ball.

Explain why the friction force must be directed uphill. $($b$)$ What is

the acceleration of the center of mass of the ball? $($c$)$ What minimum

coefficient of static friction is needed to prevent slipping?

$10\mathrm{.}31$cdots A size$-5$ football of diameter $22\mathrm{.}6cm$ and mass $426g$ rolls

up a hill without slipping, reaching a maximum height of $5\mathrm{.}00m$ above

the base of the hill. We can model this ball as a thin$-$walled hollow

sphere. $($a$)$ At what rate was it rotating at the base of the hill? $($b$)$ How

much rotational kinetic energy did it have then? Neglect rolling friction

fric$-$

ccur?

cient

and assume the system's total mechanical energy is conserved.d$)$explain which force starts motion and which force prevent slipping and why?