Name: Disc Rolling Down an Incline
Uploaded: 2019-07-19
Duration: 15 min 18 s
Description: In this video, we will learn to study the motion of a disc rolling down from an incline. We will find out the acceleration of the rolling disc.

Question

Derive an expression that relates angular momentum with the angular velocity of a rigid body.

Accepted Answer

The angular momentum of an object is directly proportional to its angular velocity.

Let us consider an object consisting of $n$ particles having masses $m_{1}, m_{2}, m_{3}, . . . . . . . ., m_{n}$ with distances $r_{1}, r_{2}, r_{3}, . . . . . . . . . . . . . . . . . ., r_{n}$ respectively from a fixed axis. Also consider that the angular velocity of the object is $ω$ .

The linear velocities of each particles can be calculated as follows-

$v_{1} = r_{1} ω v_{2} = r_{2} ω v_{3} = r_{3} ω . . v_{n} = r_{n} ω$

The linear momentum of each particle is given by

$p_{1} = m_{1} v_{1} = m_{1} r_{1} ω p_{2} = m_{2} v_{2} = m_{2} r_{2} ω . . p_{n} = m_{n} v_{n} = m_{n} r_{n} ω$

The angular momentum of each particle can be written as

$L_{1} = p_{1} r_{1} = (m_{1} r_{1} ω) r_{1} = m_{1} {r_{1}}^{2} ω L_{2} = p_{2} r_{2} = (m_{2} r_{2} ω) r_{2} = m_{2} {r_{2}}^{2} ω . . L_{n} = p_{n} r_{n} = (m_{n} r_{n} ω) r_{n} = m_{n} {r_{n}}^{2} ω$

Thus, the total angular momentum of the object can be calculated as follows-

$\begin{array}{rcl} L & = & L_{1} + L_{2} + L_{3} + . . . . . . . . + L_{n} \\ = & m_{1} {r_{1}}^{2} ω + m_{2} {r_{2}}^{2} ω + m_{3} {r_{3}}^{2} ω + . . . . . . + m_{n} {r_{n}}^{2} ω \\ = & (m_{1} {r_{1}}^{2} + m_{2} {r_{2}}^{2} + m_{3} {r_{3}}^{2} + . . . . . . + m_{n} {r_{n}}^{2}) ω \\ = & I ω \end{array}$

where, $I$ represents the moment of inertia of the object about the axis.

Derive an expression that relates angular momentum with the angular velocity of a rigid body.

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