Question 537: Centre of Mass and Rotational Motion

Question

The earth is assumed to be a sphere of radius ' $R$ ' and mass ' $M$ ' having period of rotation ' $T$ '. The angular momentum of earth about its axis of rotation is

Accepted Answer

The angular momentum of an object rotating about an axis is given by the formula:
$$ L = I \omega $$
where $ L $ is the angular momentum, $ I $ is the moment of inertia of the object, and $ \omega $ is the angular velocity.
For a solid sphere, the moment of inertia $ I $ about an axis through its center is:
$$ I = \frac{2}{5} M R^2 $$
where $ M $ is the mass and $ R $ is the radius of the sphere.
The angular velocity $ \omega $ can be expressed in terms of the period of rotation $ T $ as:
$$ \omega = \frac{2\pi}{T} $$
Plugging these expressions into the angular momentum formula, we have:
$$ L = \left(\frac{2}{5} M R^2\right) \left(\frac{2\pi}{T}\right) $$
Simplifying, we get:
$$ L = \frac{4\pi M R^2}{5T} $$
Therefore, the angular momentum of the Earth about its axis of rotation is given by:
Option B
:
$$ \frac{4 \pi \mathrm{MR}^2}{5 \mathrm{~T}} $$

Answer

$\frac{2 \pi \mathrm{MR}^2}{5 \mathrm{~T}}$

Answer

$\frac{4 \pi \mathrm{MR}^2}{5 \mathrm{~T}}$

Answer

$\frac{\mathrm{MR}^2 \mathrm{~T}}{2 \pi}$

Answer

$\frac{\mathrm{MR}^2 \mathrm{~T}}{4 \pi}$

Centre of Mass and Rotational Motion

The earth is assumed to be a sphere of radius ' R ' and mass ' M ' having period of rotation ' T '. The angular momentum of earth about its axis of rotation is

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