Relationship between Angular Momentum and Moment of Inertia | Physics Grade XI

Angular Momentum and Moment of Inertia

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Relationship between Angular Momentum and Moment of Inertia | Physics Grade XI

Relationship between Angular Momentum and Moment of Inertia

Angular momentum
Angular momentum is defined as the moment of linear momentum of an object. It is denoted by L.
 
According to definition,
L = p. r
L = m. v. r
L = m. r. ω. r
L = m. r2. ω   is the expression for angular momentum of the body
 
Relationship between Angular momentum and moment of inertia
Let a body is mass ‘M’ made up of n-particles rotate about the axis YY'. Let us suppose small particles m1, m2, m3, ……, mn and the distance between YY' and distance between YY' and it be r1, r2, r3, …... respectively.                       

Relationship between angular momentum and moment of inertia

We know angular velocity of all the particles are same but linear velocity is different due to the different position of particles. Then, Linear velocity of m1 is given by
     v1 = r1ω
 
Similarly,
     v2 = r2ω
     v3 = r3ω
 
The angular momentum of mass m1 is, L = Linear momentum * r1
L = m1*v1*r1
L = m1*r12
 
Similarly, the angular momentum of masses m2, m3, …..,mn are m2*r22*ω, m3*r32*ω, mn*rn2*ω,
The total angular momentum of the body is L = m1r12ω + m2r22ω + m3r32ω + …………... + mnrn2ω
      L = ω (m1r12 + m2r22 + m3r32 + …………... + mnrn2)
      L = I. ω

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