Answer
(a)
$v(0)$ = $0.925$ $cm/s$
$v(0.005)$ = $0.694$ $cm/s$
$v(0.01)$ = $0$ $cm/s$
(b)
$v'(0)$ = $0$ $(cm/s)/cm$
$v'(0.005)$ = $-92.592$ $(cm/s)/cm$
$v'(0.01)$ = $-185.185$ $(cm/s)/cm$
(c)
The velocity is greatest where $r$ = 0 (at the center) and the velocity is changing most where $r$ = $R$ = $0.01$ cm
(at the edge).
Work Step by Step
(a)
$v$ = $\frac{P}{4nl}(R^{2}-r^{2})$ with $R$ = $0.01$, $l$ = $3$, $P$ = $3000$ and $n$ = $0.027$
$v(r)$ = $\frac{3000}{4(0.027)(3)}(0.01^{2}-r^{2})$
$v(0)$ = $0.925$ $cm/s$
$v(0.005)$ = $0.694$ $cm/s$
$v(0.01)$ = $0$ $cm/s$
(b)
$v(r)$ = $\frac{P}{4nl}(R^{2}-r^{2})$
$v'(r)$ = $\frac{P}{4nl}(-2r)$ = $-\frac{Pr}{2nl}$
with $l$ = $3$, $P$ = $3000$ and $n$ = $0.027$
$v'(r)$ = $-\frac{3000r}{2(0.027)(3)}$
$v'(0)$ = $0$ $(cm/s)/cm$
$v'(0.005)$ = $-92.592$ $(cm/s)/cm$
$v'(0.01)$ = $-185.185$ $(cm/s)/cm$
(c)
The velocity is greatest where $r$ = 0 (at the center) and the velocity is changing most where $r$ = $R$ = $0.01$ cm
(at the edge).