Angular Momentum The angular momentum of a rigid

Angular Momentum

The angular momentum of a rigid object depends on its moment of inertia and angular velocity. The angular momentum L of a particle about an axis is defined as the moment of momentum.

A particle of mass m rotates at ω rad s-1 about the point O as shown. v r m ω O The linear momentum p = m v. The moment of p = m v r (r is perpendicular to v). Thus the angular momentum of this particle, L = m v r = m r 2 ω (since v = r ω)

For a rigid object about a fixed axis the angular momentum L is given by ∑ (mr 2ω). This can be written as ω ∑ (mr 2) since all the individual parts of the object will have the same angular velocity ω. Since I = ∑ (mr 2) the angular momentum of a rigid body is: L=Iω (unit of L: kgm 2 s-1)

Conservation of Angular Momentum In the absence of external torques, the angular momentum of a rotating rigid object is conserved. i. e. The total angular momentum before an impact will equal the total angular momentum after impact providing no external torques are acting. (I ω)before = (I ω)after

Example A girl of mass 45 Kg stands 3 m from the centre of a large playgroundabout rotating at 2 rads-1. If she moves into a radius of 1 m calculate the final angular velocity of the system. (Iroundabout= 600 kgm 2) (The girl is considered a point mass).

Before: I total = Igirl + I roundabout = mr 2 + 600 (r =3 m) I total = 45 x 32 +600 = 1005 kgm 2 After: I total = Igirl + I roundabout = mr 2 + 600 (r=1 m) I total = 45 x 12 +600 = 645 kgm 2 (I ω)before = (I ω)after 1005 x 2 =645 x ωafter = 3. 1 rads-1