A wheel with radius $40$ cm is rotating at a rate of $19$ rev$/$s$.$

$($a$)$ What is the angular speed in radians per second?

rad$/$s

$($b$)$ In a time interval of $8$ s$,$ what is the angle in radians through which the wheel rotates?

rad

$($c$)$ At

t $=10$ s

the angular speed begins to increase at a rate of $1\mathrm{.}0$ rad$/$s$/$s$.$ At

t $=15$ s$,$

what is the angular speed in radians per second?

rad$/$s

$($d$)$ Through what angle in radians did the wheel rotate during the time between

t $=10$ s

and

t $=15$ s$?$

rad

$($e$)$ If the wheel rolls along the ground without slipping, the instantaneous velocity of the atoms of the object that are momentarily in contact with the ground is zero. This zero$-$velocity condition implies that

vCM $=$R$,$

where $$ is the angular speed of the object, since the instantaneous speed of the contact point is

vCM $$R$.$

During the time between

t $=10$ s

and

t $=15$ s$,$

how far did the center of the wheel move, in meters?

m