Angular momentum Definition , Formula ,Calculation

Angular momentum

  • In any Inertial frame of Reference 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 momentum.
  • 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 momentum 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 angular momentum of the particle is given by
    L = r×p = m(r×v)               (1)
  • Angular momentum is a vector quantity and its direction is perpendicular 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

    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 perpendicular 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

    angular momentum
  • Writing angular momentum in component form we get

    angular momentum
    writing equation 5 again we get

    angular momentum
    Comparing unit vectors on both the sides we get

    angular momentum
  • 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

    Relation between angular momentum and torque
  • But from newton's second law of motion we have

    Relation between angular momentum and torque
    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

First case

Second case

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

Now rsinθ=perpendicular distance from x axis which is constant
So Angular momentum is constant

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