Rotational Motion

ROTATIONAL DYNAMICS

1.  RIGID BODY:

Rigid body is defined as a system of particles in which distance between each pair of particles remains constant (with respect to time) that means the shape and size do not change, during the motion.

Eg. Fan, Pen, Table, stone and so on.

Our body is not a rigid body, two blocks with a spring attached between them is also not a rigid body. For every pair of particles in a rigid body, there is no velocity of separation or approach between the particles. In the figure shown velocities of A and B with respect to ground are VA and VB respectively.

If the above body is rigid

V_A cos θ₁ = V_B cos θ₂

Note:

With respect to any particle of rigid body the motion of any other particle of that rigid body is circular.

V_BA = relative velocity of B with respect to A.

Pure Translational Motion:
A body is said to be in pure translational motion if the displacement of each particle is same during any time interval however small or large. In this motion all the particles have same s, v & a at an instant.

Pure Rotational Motion:
A body is said to be in pure rotational motion if the perpendicular distance of each particle remains constant from a fixed line or point and do not move parallel to the line, and that line is known as axis of rotation. In this motion all the particles have same

θ, ω and α at an instant.

Combined translation and rotational Motion
A body is said to be in translation and rotational motion if all the particles rotates about an axis of rotation and the axis of rotation moves with respect to the ground.

MOMENT OF INERTIA
Like the centre of mass, the moment of inertia is a property of an object that is related to its mass distribution. The moment of inertia (denoted by I) is an important quantity in the study of system of particles that are rotating. The role of the moment of inertia in the study of rotational motion is analogous to that of mass in the study of linear motion. Moment of inertia gives a measurement of the resistance of a body to a change in its rotational motion. If a body is at rest, the larger the moment of inertia of a body the more difficult it is to put hat body into rotational motion. Similarly, the larger the moment of inertia of a body, the more difficult to stop its rotational motion. The moment of inertia is calculated about some axis (usually the rotational axis).

Moment of inertia depends on:

1. Density of the material of body
2. Shape & size of body
3. Axis of rotation in totality we can say that it depends upon distribution of mass relative to axis of rotation.

CENTRE OF MASS:
Every physical system has associated with it a certain point whose motion characterises the motion of the whole system. When the system moves under some external forces, then this point moves as if the entire mass of the system is concentrated at this point and also the external force is applied at this point for translational motion. This point is called the centre of mass of the system.

Position of COM of two particles:-
Consider two particles of masses m₁ and m₂ separated by a distance l as shown in figure.

Let us assume that m₁ is placed at origin and m₂, is placed at position (l, 0) and the distance of centre of mass from nil & m₂ is r₁ & r₂, respectively.

From the above discussion, we see that

 if m₁ = m₂ , i.e., COM lies midway

Between the two particles of equal masses.

Similarly, r₁ > r₂ if m₁< m₂ and r₁ < r₂ if m₂ < mi i.e., COM is nearer to the particle having larger mass.

From equation (1) & (2)

_m1r₁= m₂r₂

Centre of mass of two particle system lie on the line joining the centre of mass of two particle system.