Mass moment of inertia of circular plate

Definition:
Mass moment of inertia is the resistance of an object to rotational motion around an axis.

Formula:
The mass moment of inertia of a circular plate with mass 'M' and radius 'r' is given by the formula:

I = (Mr²)/4

Explanation:

The circular plate can be considered as a 2D object with uniform mass distribution. The mass moment of inertia of a 2D object is given by the formula:

I = ∫r²dm

where 'r' is the distance of a small element of mass 'dm' from the axis of rotation.

For a circular plate, the mass distribution is uniform and the distance of each small element of mass from the axis of rotation is equal to the radius 'r'. Therefore, we can replace 'r' with 'R' which is the radius of the circular plate.

The mass of the circular plate is given by 'M' and the area is given by 'πR²'. Therefore, the mass per unit area is given by 'M/πR²'. Using this, we can express 'dm' in terms of 'dA' which is a small area element.

dm = (M/πR²)dA

Substituting this in the equation for mass moment of inertia, we get:

I = ∫R²(M/πR²)dA

I = (M/π)∫R²dA

I = (M/π)πR²

I = (Mr²)/4

Therefore, the mass moment of inertia of a circular plate with mass 'M' and radius 'r' is given by the formula:

I = (Mr²)/4

Answer:
Hence, option 'C' is the correct answer which is (Mr²)/4.