Relation Between Torque and Angular Momentum

by • 04/07/2012 • GeneralComments (0)327

Relation Between Torque and Angular Momentum

OR

Prove that the rate of change of angular momentum is equal to the external torque acting on the body.
Proof

We know that rate of change of linear momentum is equal to the applied force.

F = dP / dt

Taking cross product with r on both sides, we get

R x F = r x dP / dt

τ = r x dP / dt ……………………….. {since r x P = τ}

Now, according to the definition of angular momentum

ρ = r x P

Taking derivative w.r.t time, we get

dρ / dt = d / dt (r x P)

=> dρ / dt = r x dP / dt + dr / dt x P

=> dρ / dt = τ + V x P ……………… {since dr / dt = V}

=> dρ / dt = τ + V x mV

=> dρ / dt = τ + m (V x V)

=> dρ / dt = τ + 0 …………….. {since V x V = 0}

=> dρ / dt = τ

Or, Rate of change of Angular Momentum = External Torque …………….. (Proved)

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