Dynamics of Rigid Bodies formulae for NEET

The rigid body is an idealization of a solid body where the deformations occurring on the body are neglected. In other words, the distance between any two given points of a rigid body remains a constant regardless of the external force acting upon it. The two types of motion a rigid body can undergo are;

  • Translational Motion
  • Rotational Motion

Moment of Inertia

Moment of Inertia is defined as the capacity of the system to oppose the change produced in the rotational motion of the body.

For a single particle moment of inertia I=mr2

Here m is the mass of the particle and r is the perpendicular distance from the axis about which moment of inertia is to be calculated.

For many particles, I = miri2

Moment of Inertia of different objects

moment of inertia for different objects

Radius of Gyration

K = √I/M

Here

M is the mass of the rotating object

I is the moment of inertia

Relation Between Torque and Moment of Inertia

Relation Between Torque and Moment of Inertia

τ is the torque (twisting effect of force)

I is the moment of inertia

α is the angular acceleration ( the rate of change of angular velocity)

Angular momentum

Angular momentum

Point object:  For an object accelerating around a fixed point. For example, Earth revolving around the sun. Here the angular momentum is given by:

\(\vec{L} = r \times \vec{p}\)

L is the angular velocity

r is the radius

p is the linear momentum

Extended object: The object, which is rotating about a fixed point. For example, Earth rotating about its axis. Here the angular momentum is given by

\(\vec{L} = r \times \vec{\omega }\)

ω is the angular velocity

L = angular momentum of the object

I = Moment of inertia

Relation between angular momentum and Torque

torque and angular momentum relation

Torque is the change in angular momentum