Question

Derive the formula of total kinetic energy of a body at the bottom rolling down on an inclined plane.

Answer 1

Relation between Angular Momentum, Moment of Inertia and Angular Velocity

We know that the angular momentum of a body is given as;

$\overrightarrow{J}=\overrightarrow{r}\times \overrightarrow{p}$

or $J=rpsin\theta \hat{n}$

If $\overrightarrow{r}$ and $\overrightarrow{p}$ are perpendicular, then $\theta ={90}^{\circ }$

$sin{90}^{\circ }=1$

$J=rp$

$=rmv$

$=rm\left(r\omega \right)(\because v=r\omega )$

$J=m{r}^2\omega$

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 ais

$J=\Sigma m{r}^2\omega$

$\because \omega$ is constant, hence $J=\omega \Sigma m{r}^2$

$J=I\omega (I=\Sigma m{r}^2)$ .......(1)

If $\omega =1$ rad/sec 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.