ROTATIONAL MOTION
5.1 Rotational Motion of a rigid body about a fixed axis

A rigid body is defined as a system of particles, the distance between any pair of which remains unaltered. This is an ideal condition, but objects like a metal block or a log of wood under moderate forces and torques approach very close to being 'rigid' bodies.

The rotational motion of a rigid body about a fixed axis can be described by introducing concepts of Moment of Inertia, torque, angular momentum etc.

Consider a rigid body rotating about a fixed axis. Let m_i, be the mass of the i^th particle, let r_i be its^^r distance from the axis and let vi be its tangential velocity ^^r to radius r_i of the circle in which the particle rotates with same angular velocity as that of the body.

The Quantity I is called the moment of Inertia (M.I) of the rigid body about the given axis. It is defined as the sum over the body of mass of each particle multiplied by the square of its ^^r distance from the axis of rotation. Its unit is 1 Kg x m².

Torque on the rigid body

Angular Momentum of the Rigid Body

Significance of Moment of Inertia

Consider Equations (1) , (4) , (6) as above viz,

These equations are analogous to corresponding equations for a particle in translational motion.

analogous to 'm' ; hence the name Moment of Inertia (M.I). It is the inertia of a rigid body in rotational motion about a fixed axis, i.e., the rigid body cannot change its state of rotational motion about a given axis by itself due to its M.I.

The similarity pointed out above between translational motion of a particle and rotational motion of a rigid body can be extended to any equation for the translational motion to the obtain corresponding rotational motion equations,

The table below lists these similarities.

Note : For a discrete system of particles the where as for a continuous system of particles I =ò dm r²

M.I of Some regular Solids

The list below gives the expressions for M.I (calculated on the basis of I =òdm r² formula) for some regular solids:

Rectangular Block : Mass M; Length L, Breadth B, Height H; Axes^^r to face and through center.

Ring: Mass M ; Radius r, Axis : Geometrical Axis
I = MR²                    

Radius of Gyration : K

The perpendicular distance from the given axis, at which the whole mass m of the body can be imagined to be concentrated is called K, such that I = MK² . This K then is called radius of Gyration.

 

Rolling Motion

When a rigid body rotates about an axis, and the axis itself undergoes translational motion then the combined motion of the rigid body is called rolling motion.

Consider the rigid body of a circular cross- section as shown, with an instantaneous angular velocity 'w' and an instantaneous linear velocity of the center of the mass of  the body 'v';  then   v = Rw and similarly a = ra.

Law of Conservation of angular momentum

Total angular momentum of a system which is isolated, so that no external torque acts on the system, is always conserved.

This follows from the law of motion.

In many daily life phenomena the processes are governed due to this law.

1. A ballet dancer increases or decreases his/her spin motion speed by folding his/her arms as close to his/her body or by spreading his/her arms away from his/her body respectively.
2. A diver folds his/her body in as round a shape as tightly as possible to perform a greater number of somersaults before striking the water.

Solved Problems
1. Does force of friction always oppose motion ? Can it be a cause of motion, explain with examples.

Solution

The force of friction does not necessarily always oppose motion ; one kind of force of friction can produce, i.e. can be the cause of change in state of motion but simultaneously some other kind of force of friction will oppose the motion though with lesser effect.

For example,

1. While negotiating bends in a road the vehicle through its wheels pushes the (non-slippery) road in the outward direction, the road then exerts static force of friction in the inward direction which can provide the centripetal force. However the force the kinetic friction or rather rolling friction continues to oppose the motion along tangential direction.
2. An object like ring, disc, cylinder, sphere cannot be rolled over a surface unless the surface is rough. The static force of friction then provides the necessary turning moment ; once the rolling motion starts, then the force of rolling friction continues to oppose the rolling motion.

2. If a ring, a disc, a sphere all of the same mass and radius initially at rest on top of an inclined plane (rough are released then the order in which they arrive earliest, later and latest is sphere, disc, ring respectively, why is this so, explain analytically.

Solution

\ Center of mass (C.M.) of a rolling body moves effectively with uniform translational acceleration

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