Rotational dynamics of rigid bodies: Angular momentum, Torque

ROTATIONAL DYNAMICS OF RIGID BODIES

Angular momentum

The moment of linear momentum is known as angular momentum.

Consider a particle of mass m at a distance r from the axis of rotation. Let v be the linear velocity of the particle (Fig. 1.37). Then,

Angular momentum

= linear momentum × distance

where ω is the angular velocity of the particle. 

Also =

S.I unit for angular momentum is kgm2 s-1

Definition

Angular momentum of a particle is defined as its moment of linear momentum. It is given by the product of linear momentum and perpendicular distance of its line of action from the axis of rotation. It is denoted by

It is a vector quantity. The direction of angular momentum is given by right hand rule. According to this rule, if the fingers of right hand are curled in the direction of rotation about the axis, then the thumb points on the direction of angular momentum.

In vector notation, angular momentum is given as the vector product of and the linear momentum i.e., 

The direction of angular momentum is perpendicular to plane containing

Expression for Angular momentum of a rigid body

Consider a rigid body rotating about a fixed axis XOX'. The rigid body consists of a large number of particles.

Let m1, m2, m3, ... etc. be the masses of the particles situated at distances r1, r2, r3, ... etc. from the fixed axis.

All the particles rotate with the same angular velocity ω. (Fig. 1.39)

Angular momentum

= linear momentum × distance

= mv × r

= mr ω × r = mr ω × r

= mr2 ω        { ⸪ v = r ω}

∴ Angular momentum of the first particle = m1 r12 ω

Angular momentum of the second particle = m2 r22 ω

Angular momentum of the third particle = m3 r32 ω

and so on.

The angular momentum of the whole body is equal to the sum of the angular momentum of all the particles of the body.

The angular momentum of the rigid body  = m1 r12 ω + m2 r22 ω + m3 r32 ω

= ω (m1 r12 + m2 r22 + m3 r32 + ...) 

= ω Σ mr2

Let I be the moment of inertia of the rigid body about the fixed axis, I = Σ mr2

∴ The angular momentum of the rigid body, = ω I 

Torque (τ)

Torque is the turning effect of a force on a body, on which the force acts.

The turning effect of a force depends on 

(i) the magnitude of the force and 

(ii) the perpendicular distance from the axis of rotation to the line of action of the force

Definition

The moment of the applied force is called torque. It is represented by the symbol ‘τ’.

If F is the force acting on a body at a distance r (Fig. 1.40) then,

Torque = Force × distance

i.e., τ = F × r

The rotational motion comes into picture only when the torque acts on the body. 

Torque in vector notation

When a force is applied on a rigid body capable of rotation about some axis, the body rotates about the axis.

The ability of a force to rotate a body about an axis is measured in terms of a quantity called torque.

Consider a body capable of rotation about an axis passing through O. Let a force F act at A, distance 'r’ from O such that the line of action of the force is perpendicular to OA.

The moment of this force F about the axis through O is a measure of the torque. (Fig 1.40)

If the direction of the force is inclined at an angle θ with the direction of r, it is measured as the product of the force F and the perpendicular distance from the axis of rotation to the line of action of the force.

∴ Torque = F × ON = F × d = F.r sin θ

This can be expressed in vector form as

Thus torque is the cross product of force and the vector between the axis of rotation and the point of application of the force.

It is acting in a direction perpendicular to the plane containing and its direction is given by right hand screw rule.

If perpendicular to each other τ = rF

(⸪ θ = 90° and sin θ = 1)