RIGID BODY DYNAMICS

Dynamics is the branch of mechanics that describes motion of any object.

Dynamics of a point particle is easier to study than a rigid body. We take help of vectors to deal with both the cases.

To describe the motion of any object, it is necessary to know its position vector and velocity, with the help of which we can predict its motion.

In most of the problems, we use Cartesian coordinates to describe particle�s trajectory. Frame of reference plays an important role in dynamics problems.

RIGID BODY:

A rigid body is one in which the distance between any two particles remains a constant. However, no body is perfectly rigid in reality.

There are five kinds of motion associated with a solid object:

Translation
Rotation about stationary axis
Motion in a plane (2D)
Motion about a point
Free motion.

Translation and rotation about a stationary point are basic kinds of motion.

Translation:

In this kind of motion, any straight line fixed to the solid object remains parallel to its initial orientation all the time. All points in the object travel equal distances in the same time interval. Hence, at a particular instant of time,, its velocity and acceleration are same at all points.

Rotation about a stationary point:

In this kind of motion, every point on the solid object undergoes an infinitesimal amount of rotation d? in the interval dt about a stationary axis which is also called axis of rotation.

MOMENT OF FORCE:

When an external force acts on a body, it tends to rotate it about an axis. When it does so, it is said to exert torque on the body. The magnitude of torque being equal to the product of force and perpendicular distance of line of action of force from the axis of rotation.

t=F X d

It tends to rotate the body either in clockwise or in anticlockwise direction.

If line of force passes through axis of rotation then d is zero, hence torque is also zero.

On the other hand, greater the distance between line of action of force and axis of rotation, larger is the moment of force.

ANGULAR ACCELERATION:

In rotational motion, as stated above all the particles in the body move in a circle of different radii in general but all their centers lie on the common axis of rotation.

GENERAL MOTION OF A RIGID BODY

A rigid body performs both translational motion as well as rotational motion.

Its center of mass performs pure translational motion. All the forces acting on the body can be considered to be concentrated at the center of mass of the object.

a_cm= F/M

Expression of Torque in Cartesian coordinates

The direction is given by vector cross product rule. It is perpendicular to the plane in which r and F lie.

In the above expression, torque is along Z-direction.

ANGULAR MOMENTUM:

It is defined as moment of linear momentum about a given axis.

Relation between angular and linear velocity

Linear distance travelled by a particle on an object that is rotating be S,

MOMENT OF INERTIA:

It is given by the product of mass of each particle and square of its distance from the axis of rotation. It depends upon the distribution of masses relative to the axis.

Steiner theorem:

Moment of inertia about any arbitrary axis is equal to the moment of inertia about its center of mass relative to that axis and is parallel plus the product of mass m of the body and the square of the distance between the two axes.

Perpendicular axis theorem:

Theorem states moment of inertia about an axis perpendicular to the plane is equal to the sum of moments of inertia about any two mutually perpendicular axes.

I_z=I_X+I_Y

I_X=I_y+I_z

I_Y= I_Z+I_X

Relation between torque and moment of inertia:

ROLLING MOTION

It is a special kind of motion shown by round object where specific conditions are satisfied.