Relationship between Torque and Moment of Inertia | Physics Grade XI

Torque and Moment of Inertia

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Relationship between Torque and Moment of Inertia | Physics Grade XI

Relationship between Torque and Moment of Inertia

Torque
The turning effect of force in a body is called torque or moment of force. Generally, it is denoted by τ.
     Torque = force * perpendicular distance from the axis of rotation
     or, τ = r. F

Relationship between torque and moment of inertia

relationship between torque and moment of inertiaLet a body of mass ‘M’ made of n particles rotate about an axis YY'. Let m1, m2, m3, ……. Be the mass of particles at the distance of r1, r2, r3 from the axis of rotation. As the angular velocity of all the particles are equal but the linear velocity differs.
     v1 = r1ω
 
Differentiation w.r.t to time,
     dv1/dt = dr1ω/ dt
     α1 = r1α
 
Since, f = ma,
     For m1,    f1 = m1 a = m1r1α
     For m2,    f2 = m2 a= m1r1α
     For m3,    f3 = m3 a= m1r1α

We know,
     Torque for m1, τ = r1. τ = m1r12α
     Torque for m1, τ = r2. τ = m2r22α
     Torque for m1, τ = r3. τ = m3r32α
 
Total torque,
     τ = m1r12α + m2r22α + m3r32α + ……. + mnrn2α
     τ = α (m1r12 + m2r22 + m3r32+ ……. + mnrn2)
     τ = α. I

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