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Derive the relation between torque and moment of inertia?

Answer
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Hint: TORQUE- Torque is the measure of the force that can cause an object to rotate about an axis. Force is what causes an object to accelerate in linear kinematics, similarly, torque is what causes an angular acceleration. Hence, torque can be defined as the rotational equivalent of linear force.
MOMENT OF INERTIA- The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

Step-By-Step answer:
Now we will find the relation between torque and moment of inertia-
When a torque acts on a body rotating about an axis, it produces an angular acceleration in the body.
Let, the angular velocity of each particle be ω. Then,
Angular acceleration (α)=dωdt
The linear acceleration will depend on their distance r1,r2......rn from the axis of rotation.
Consider a particle P of mass m1 at a distance r1. Let its linear velocity be v1.
Linear acceleration of 1st particle = a1=r1α
Force acting on 1st particle = F1=m1r1α
Moment of force F1 about axis of rotation is
τ1=F1r1=m1r12α
Total torque = τ=τ1+τ2+..........+τn
τ=m1r12α+m2r22α+.......+m1rn2α
τ=(m1r12+m2r22+m3r32+.......+mnrn2)α
τ=(mr2)α
τ=Iα, where I is the moment of inertia.
This is the relation between torque and moment of inertia.

NOTE- The torque produced in a body makes the body rotate about an axis, which is called the axis of rotation. In physics, torque is simply the tendency of a force to turn or twist. The formula used to calculate the torque is given by τ=F.r.sinθ. Unit of torque is Newton-meter (N-m).