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Chapter: 11th 12th std standard Class Physics sciense Higher secondary school College Notes

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Angular momentum of a rigid body

Angular momentum of a rigid body
Angular momentum of a particle : The angular momentum in a rotational motion is similar to the linear momentum in translatory motion. The linear momentum of a particle moving along a straight line is the product of its mass and linear velocity (i.e) Vec p = m.


Angular momentum of a particle

 

The angular momentum in a rotational motion is similar to the linear momentum in translatory motion. The linear momentum of a

particle moving along a straight line is the product of its mass and linear velocity (i.e) Vec p = m. Vec v. The angular momentum of a particle is

defined as the moment of linear momentum of the particle.

Let us consider a particle of mass m moving in the XY plane with a velocity v and linear momentum p =   mv  at a distance r from the origin (Fig. ).


The angular momentum L of the particle about an axis passing through O perpendicular to XY plane is defined as the cross product of Vec  r and Vec  p.

 (i.e) Vec  L = Vec  r ? Vec  P

Its magnitude is given by L = r p sin θ

 

Where θ is the angle between  Vec r and Vec  p and L is along a direction perpendicular to the plane containing  Vec  r and Vec  p .

 

The unit of angular momentum is kg m2 s?1 and its dimensional formula is, M L2 T?1.

 

Angular momentum of a rigid body

 

Let us consider a system of n particles of masses m1, m2 ?.. mn situated at distances r1, r2, ?..rn respectively from the axis of rotation (Fig. ). Let v1,v2, v 3 ?.. be the linear velocities of the particles respectively, then linear momentum of first particle = m1v1.

 

Since v1= r1ω the linear momentum of first particle = m1(r1 ω)

The moment of linear momentum of first particle

= linear momentum ? perpendicular distance

= (m1r1ω) ? r1

angular momentum of first particle = m1r12ω

Similarly,

 

angular momentum of second particle = m2r22ω

 

angular momentum of third particle = m3r32ω and so on.

 

The sum of the moment of the linear momenta of all the particles of a rotating rigid body taken together about the axis of rotation is known as angular momentum of the rigid body.

 

 Angular momentum of the rotating rigid body = sum of the angular momenta of all the particles.

(i.e)  L = m 1r1 2ω+ m2 r2 2ω + m3 r32ω.?. + mn rn 2 ω

L = ω [  m 1r1 2+ m2 r2 2ω + m3 r32.?. + mn rn 2 ]

L = ωI

where I = ∑ mi ri 2 moment of inertia of the rotating rigid body about the axis of rotation.


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