Question

A cylinder of mass 5 kg and radius 30 cm,is free to rotate about its axis;it receives an angular impulse of 3 kg square metre per seconds,initally followed by a similar impulse after every 4 s .What is the angular speed of the cylinder after 30s of the initial impulse?

Answer 1

it is given that,
mass of cylinder, m = 5kg
radius of cylinder , r = 30cm = 0.3 m
angular impulse , ∆L = 3kgm²/s

here, initial angular speed is $\omega_0=0$
angular speed after 4s = $\omega$

we know,
angular impulse = change in angular momentum
3 = $I(\omega-\omega_0)$
Here I is moment of inertia.
3 =1/2 mr²($\omega$ -0)
6 = 5 × (0.3)² × $\omega$
$\omega$ = 40/3 rad/s

now, use $\omega=\omega_0+\alpha t$
40/3 = 0 + $\alpha$ × 4
$\alpha$ = 10/3 rad/s

now angular speed after 30 sec = 0 + 10/3 × 30
= 300/3 = 100 rad/s

Answer 2

Inthis I=mr^2/2 so I=5x0.3x0.3/2

I=0.45/2

The cylinder receives 8 times angular impulse of 3 kg-m^2/s in 30 second in period of 4 sec. so total will be 24 unit.

L=I×omega

Omega=24x2/0.45

Omega=106.7 rad/s.

Hence answer