Question

A homogeneous disc of mass 1 kg and radius 15 cm is rotating with angular velocity of 2 rad/s. The linear moment of disc about given axis
A. 2.4 kg m/s
B. 0.3 kg m/s
C. 4 kg m/s
D. Zero

Answer 1

Hint: To solve this question, use the formula for angular momentum in terms of moment of inertia and angular velocity. Then calculate the moment of inertia of the disk. Substitute the values in the formula for angular momentum. Now, use the relation between angular momentum and linear momentum, rearrange the terms and then substitute the values in the obtained expression. This will give the linear moment of disc about a given axis.
Formula used:
$L=I \Omega$
$I=\dfrac {1}{2}M{R}^{2}$
L=pR

Complete answer:
Given: Mass of the disc (M): 1 kg
           Radius of the disc (R): 15 cm= 0.15 m
           Angular velocity $\left (\Omega \right)$: 2 rad/s
Angular momentum is given by,
$L=I \Omega$ …(1)
Where, I is the Moment of inertia
$\Omega$ is the angular velocity
We know, moment of inertia for a disk is given by,
$I=\dfrac {1}{2}M{R}^{2}$
Substituting values in above equation we get,
$I=\dfrac {1}{2} \times 1 \times {\left(0.15 \right )}^{2}$
$\therefore I= \dfrac {1}{2} \times 0.0225$
$\therefore I=0.01125 kg {m}^{2}$
Now, substituting values in equation. (1) we get,
$L=0.01125 \times 2$
$\therefore L= 0.0225 kg {m}^{2} rad/s$
We know, the relation between angular momentum and linear momentum is given by,
L=pR …(2)
Where, p is the linear momentum
Rearranging equation.(2) we get,
$p= \dfrac {L}{R}$
Substituting values in above equation we get,
$p=\dfrac {0.0225}{0.15}$
$\therefore p=0.15 kg \quad {m}/{s}$
Thus, the linear moment of disc about the given axis is 0.15 kg m/s.
So, the calculated answer does not match any given options.

Note:
To answer these types of questions, students must know the required formulas. They should not get confused between the formula for moment of inertia for disc and that for a ring. The moment of inertia is different for a ring and disc. Ring and disc have different moments of inertia as the disc has a solid volume whereas the ring only has a perimeter.