Question

Establish the relation between moment of inertia, torque, angular acceleration and define moment of inertia on the basis of this relation.

Answer 1

Suppose a rigid body is made of n particles of masses m₁, m₂, m₃, ......, m_n and their distances from the axes of rotation YY' are r₁, r₂, r₃, .........., r_n respectively. If a₁, a₂, a₃, ......., a_n are linear accelerations of the particles, then forces acting on the particles,

If the body is rotating with the angular acceleration α, then

F₁ = m₁a = m₁r₁α (because a₁ = r₁α)

∵ Torque = force x normal distance from the axis of rotation

or τ = F x r

or τ = mrα x r = mr²α

∴ τ₁ = m₁r₁²α; τ₂ = m₂r₂²α; ....., τ_n = m_nr_n²α

∴ If all these torque are in the same direction then net torque acting on the body,

τ = τ₁ + τ₂ + τ₃ + .......+ τ_n

= m₁r₁²α + m₂r₂²α + m₃r₃²α + ..... + m_nr_n²α

= (m₁r₁² + m₂r₂² + m₃r₃² + ........ + m_nr_n²)α

\( = \sum_{i = 1}^n\ m_ir_i^2\ .\ \alpha\)

or \(\tau = I\alpha\) 

i.e., Torque = Moment of inertia x Angular acceleration,

If α = 1 rad.s^-1, then τ = I

i.e., "Momentum of inertia of a body about a given axis of rotation is equivalent to torque which can produce unit angular acceleration in the body."