First slide

Rotational motion

Question

Figure shows a plank of mass m moving at speed u on a smooth surface. A spinning ball (sphere) of mass m and radius r normally strikes it at speed v. The friction coefficient between ball and upper surface of plank is $μ$ = 0.5 and coefficient of restitution is e = 0.5. Find the angular velocity of sphere just after collision.

Moderate

A

$\frac{12}{4} \frac{μ v}{r}$
B

$\frac{15}{4} \frac{μ v}{r}$
C

$\frac{15}{6} \frac{μ}{r}$
D

$\frac{15}{4} \frac{v}{r}$

Solution

As shown in figure if we write impulse = change in momentum
equations $mu - J_{2} = {mv}_{2}$
$mv - J_{1} = - \frac{mv}{2} \Rightarrow J_{1} = \frac{3}{2} mv$
$0 + J_{2} = {mv}_{1} \Rightarrow v_{1} = \frac{3}{2} μ v = \frac{3}{4} v$
velocity of ball $v_{ball} = \sqrt{(v / 2)^{2} + (3 v / 4)^{2}} = \frac{\sqrt{13}}{4} v$
we also have $\frac{2}{5} {mr}^{2} ω - J_{2} r = \frac{2}{5} {mr}^{2} ω_{1}$
$\frac{2}{5} m r^{2} ω - μ (\frac{3}{2} mv) r = = \frac{2}{5} {mr}^{2} ω_{1}$
$ω_{1} = \frac{15}{4} \frac{μ v}{r}$

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