Relation between Angular Momentum, Moment of Inertia and Angular Velocity
We know that the angular momentum of a body is given as;
\(\vec{J}=\vec{r} \times \vec{p}\)
or J = r p sinθ \(\hat{n}\)
If \(\vec{r}\) and \(\vec{p}\) are perpendicular, then θ = 90°
sin 90° = 1
J = r p
= r m v
= r m(r ω) (∵ v = rω)
J = mr2ω
The angular momentum of the body will be equal to the sum of all the moments of linear moment of the particles i.e., angular momentum relative to the rotational axis
J = Σ mr2 ω
∵ ω is constant, hence J = ω Σ mr2
J = Iω (I = Σ mr2) …………… (1)
If ω = 1 rad/s then I = J
Hence, the moment of inertia of a body about the axis is equal to the angular momentum if it is rotating with unit angular velocity.
