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What is the formula for Torque acting on a body?

Torque is defined as the rate of change of angular momentum of an object. The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing.
The formula for Torque acting on a body is given as,

Torque (τ) = moment of inertia (I) × angular acceleration (α)
→             →
τ     = Iα  

where τ = Torque
α = angular acceleration
I = moment of inertia
S.I. unit of torque is Newton metre (Nm)

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What is the formula for kinetic energy of rotating rigid body?

The kinetic energy of the whole body is equal to the sum of the kinetic energy of all the particles present in the body. A rotating rigid body has kinetic energy because all atoms in the object are in motion.

The formula for kinetic energy of a rotating rigid body is given as,
Kinetic Energy (K.E.) = 1 /2 ( Moment of inertia (I) × (angular velocity)2)

kinetic energy of rotating rigid body

where  ω = angular velocity

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What is the formula for Radius of gyration?

Radius of gyration is the distance between the given axis and the centre of mass of the body. The centre of mass of a body is point where the entire mass of the body is supposed to be concentrated. It is denoted by ‘K ’.

If M is mass of the body, the moment of inertia is given as,
Moment of inertia (I) = mass of the body (M) × (radius of gyration)2

So, the formula for radius of gyration (K) is given as,

radius of gyration

where K = radius of gyration, I = Moment of inertia and M = mass of the body.

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What is the formula for Moment of Inertia of a rigid body?

Inertia of a body is its inability to change by itself its state of rest or of uniform motion in a straight line. Similarly Moment of inertia of a body is its inability to change by itself its state of rest or of uniform rotatory motion about an axis.
The formula for Moment of Inertia of a rigid body is given as,
The moment of inertia of the rigid body (l) = Sum of moments of inertia of
all the particles present in the body.
Let us consider,

m1, m2, m3 = masses of large particles in a rigid body

r1, r2, r3 = distance of the masses from the axis
The formula for Moment of Inertia of a rigid body will be,

 l= m1r12 + m2r22 + m3r32 + …

The above formula can also be expressed as,

 Moment of Inertia of a rigid body

mi = mass of ith particle

ri = distance between ith particle and axis o rotation

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What is the formula for Moment of Inertia of a particle?

Moment of inertia of a particle about an axis is equal to the product of the mass of the particle and square of its distance from the axis.

The formula for Moment of Inertia of a particle is given as,
Moment of inertia of the particle about the axis = mass of the particle (m) × distance of particle (r)
from the fixed axis

Formula for Moment of inertia of the particle about the axis =  mr2
The S.I. unit for moment of inertia is kg m2

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