Rotational Mechanics

 ROTATIONAL MECHANICS

• When a rigid body rotates about a fixed axis every particle of the body moves in a circular path.
• The perpendicular distance from axis of rotation to given point is called radius vector.
• In rotation of a body about a fixed axis angular variables are the same for all particles but linear variable changes.
• Relation between angular and linear variables.

1. V = rω
2. a = rα

Kinematical Equation of rotatory Motion

When ω0 : Initial angular velocity

ω : Final angular velocity

α : Uniform angular acceleration

t : time

Torque:

The turning effect of a force about the axis of rotation is called the moment of force or torque. Torque measures the force that can cause an object to rotate around an axis. Since the force causes the object to accelerate in linear kinematics, torque causes the object to achieve angular acceleration.

It is a vector quantity. The torque vector's direction depends on the force's direction on the axis.

Torque can be static or dynamic

Static torque does not produce angular acceleration. Someone who pushes a closed-door by using torque from the door because the door does not turn on its hinges, despite the force exerted. A person who rides a bicycle at a constant speed and uses static torque because he is not fast.

The drive shaft of a speeding car from its start has a dynamic torque as it must produce an angular acceleration of the wheels when considered to be accelerating a vehicle on the track.

The words used to describe torque can be confusing. Engineers sometimes use the word or moment of power in exchange for torque. The radius at which energy works is sometimes called a temporary arm.

Torque = Force × Perpendicular distance of line of action of force from axis of rotation.

Couple:

Two forces equal in magnitude but opposite in direction acting at two different points of a body constitute coupe.

Moment of Couple:

The product of magnitude of force in couple and perpendicular distance between them.

Moment of Inertia:

Inability of a body to change its state of rotation by itself is called moment of inertia (I). Moment of inertia is analogue to mass in translatory motion.

Mathematical Definition:

Moment of inertia of a rigid body about a given axis of rotation is the sum of products of the masses of various particles and square of their perpendicular distance from the axis of rotation

 

Moment of intertia of different bodies:

1. Rod – axis Passing through its centre and perpendicular to its length Ml2/12.
2. Rectangular plate or bar – axis Passing through its centre and perpendicular to plane Ml2/12 + Mb2/12.
3. Circular disc – axis Passign through its centre and perpendicular to its plane or transverse axis MR2/2.
4. Solid cylinder – Its own axis MR2/2.

Radius of gyration:

It is defined as the distance from the axis of rotation to a point where whole mass of the rotating body supposed to be concentrated. It is denoted by ‘K’. If ‘K’ is radius of gyration. Then I = mK2.

Important Points:

1. The moment of inertia of a body depends on the mass of the body and its distribution about the axis of rotation.
2. Moment of inertia changes with change in position of axis of rotation.
3. Radius of gyration is not a concstant quantity. Its value changes with change in location of axis of rotation.