Question

Calculate the moment of inertia of a ring along an axis passing through the center and perpendicular to the plane.

Answer 1

Moment of inertia of a ring about an axis passing through its center and perpendicular to its plane : Suppose a ring of mass M and radius R is placed in X- Y plane with its center at the origin of cartesian system. Z-axis will be our required axis in the question.

The mass of the ring is uniformly distributed to its whole circumference. Therefore, its whole circumference can be supposed to be made up of n parts of masses respectively m₁, m₂, m₃, …………., m_n.
Each mass element is situated at the same distance R from the Z-axis (i.e, schedued axis). Therefore the moment inertia of the ring,
I = I₁ + I₂ + I₃ + ……. +I_n
= m₁R² + m₂R² + m₃R² + …………… + m_nR²
= (m₁ + m₂ + m₃ + ……….. + m_n )R²
or I = MR²
where M = mass of ring.