The force acting on a particle of mass m1 situated at A, at a distance r1, from the axis of rotation = mass ? acceleration

*Relation
between torque and angular acceleration*

Let us consider a rigid body rotating about a
fixed axis X0X′ with angular velocity *ω* (Fig.).

The force acting on a particle of mass m_{1}
situated at A, at a distance *r*_{1},
from the axis of rotation = mass ? acceleration

= m_{1} ? d/dt(r_{1}ω)

= (m_{1} r_{1})dω /dt

= (m_{1} r_{1})d^{2}θ/dt^{2}

The moment of this force about
the axis of rotation

= Force ? perpendicular
distance

= (m_{1} r_{1})d^{2}θ/dt^{2
} x r_{1}

Therefore, the total moment
of all the forces acting on all the particles

or τ = Iα

where ∑ m_{1}r_{1}^{2}
= moment of inertia I of the rigid body and α = d^{2} θ /dt^{2}
angular acceleration.

**Relation between torque and angular momentum**

The angular momentum of a
rotating rigid body is, L = I ω

Differentiating the above
equation with respect to time,

dL/dt = I(d ω /dt) = Ia

where α = dω/dt angular
acceleration of the body.

But torque τ = Iα

Therefore, torque τ = dL/dt

Thus the rate of change of
angular momentum of a body is equal to the external torque acting upon the
body.

**Conservation of angular momentum**

The angular momentum of a
rotating rigid body is, L = I ω

The torque acting on a rigid
body is, τ = dL/dt

When no external torque acts on the system, τ =dL/ * _{dt}* = 0

(i.e) *L = I* ω = constant

Total angular momentum of the body = constant

*(i.e.)
when no external torque acts on the body, the net angular momentum of a
rotating rigid body remains constant. This is known as law of conservation of
angular momentum.*

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