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Angular momentum




Angular momentum



  • In any inertial frame of refrance the moment of linear momentum of a particle is known as angular momentum or, angular momentum of a particle is defined as the moment of its linear momentul.
  • In rotational motion angular momentum has the same significance as linear momentum have in the linear motion of a particle.
  • Value of angular momentum of angular momentum is equal to the product of linear momentum and p(=mv) and the position vector r of the particle from origin of axis of rotation.


    Angular momentum

  • Angular mmomentum vector is usually represented by L.
  • If the linear momentum of any particle is p=mv and its position vector from any constant point be r then abgular momentum of the particle is given by
    L = r×p = m(r×v)               (1)
  • Angular momentum is a vector quantity and its direction is perpandicular to the direction of r and p and could be found out by right hand screw rule.
  • From equation 1 scalar value or magnitude of angular momentum is given as
    |L|=rpsinθ                    (2)
    where V is the angle between r and p.
  • For a particle moving in a circular path
    v=ω×r;                    (3)
    where ω is the angular velocity.


    Angular momentum and angular velocity

    Therefore
    L=m[r×(ω×r)] = m{ω(r.r)-r(r.ω)} = mr2ω=Iω;                    (4)
    (r.ω)=0 because in circular motion r and ω are perpandicular to each other. Here I is the moment of inertia of the particle about the given axis also the direction of L and ω is same and this is a axial vector.
    writing equation 1 in the component form we get


  • Writing angular momentum in component form we get


    writing equation 5 again we get


    Comparing unit vectors on both the sides we get


  • Unit of angular momentum in CGS is gm.cm2/sec and in MKS system it is Kgm.m2/sec or Joule/sec.

Relation between angular momentum and torque


  • Differentiating equation 1 w.r.t. t we get


  • But from Newton's second law of motion we have


    Hence rate of change of angular momentum with time is equal to the torque of the force.

Question 1. A mass is whirled in a circular path with constant angular velocity and its angular momentum is L.If the string is now halve keeping the angular velocity same then angular momentum is
a. L
b. L/4
c. L/2
d. 2L
Solution 1
Angular momentum for this is defined as
=mr2ω

First case
L=mr2ω

Second case
Lf=m(r/2)2ω

So, Lf=L/4

Question 2.A mass is moving with constant velocity along a line parallel to xaxis away from origin.its angular momentum with respect to origin is
a. is zero
b. remains constant
c. goes on increasing
d. goes on decreasing

Solution 2

L=(mv)Xr
or
L=mvrsinθ
Now rsinθ=perpendicular distance from x axis which is constant
So Angular momentum is constant


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