# State and prove: Law of conservation of angular momentum. - Physics

Derivation

State and prove: Law of conservation of angular momentum.

State and prove the principle of conservation of angular momentum.

State and explain the principle of conservation of angular momentum. Use a suitable illustration. Do we use it in our daily life? When?

#### Solution

Statement:

The angular momentum of a body remains constant if the resultant external torque acting on the body is zero.

I1ω1 = I2ω2 (when τ = 0)

Here I is the moment of inertia and ω is angular. velocity.

Proof:

Consider a particle of mass m, rotating about an axis with torque ‘τ’.

Let vecp be the linear momentum of the particle and vecr be its position vector.

∴ Angular momentum, vecL = vecr xx vecp .....(1)

Differentiating equation (1) with respect to time t, we get,

(dvecL)/(dt) = d/dt(vecr xx vecp) = vecr(dvecp)/(dt) + vecp(dvecr)/(dt)

We know that, (dvecp)/(dt) = vecF, (dvecr)/(dt) = vec"v", vecp = mvec"v"

∴ (dvecL)/(dt) = vecr xx vecF + m(vec"v" xx vec"v")

∴ (dvecL)/(dt) = vecr xx vecF ............(∵ vec"v" xx vec"v" = 0)

∴ (dvecL)/(dt) = vectau.............(∵ vecr xx vecF = vectau)

Now, If the vectau = 0, then

(dvecL)/(dt) = 0

∴ vecL is constant. Hence angular momentum remains conserved.

Example:

An athlete diving off a high springboard can bring his legs and hands close to the body and performs Somersault about a horizontal axis passing through his body in the air before reaching the water below it. During the fall his angular momentum remains constant.

Concept: Conservation of Angular Momentum
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Chapter 1: Rotational Dynamics - Very Short Answer

#### APPEARS IN

Balbharati Physics 12th Standard HSC Maharashtra State Board
Chapter 1 Rotational Dynamics
Exercises | Q 9 | Page 24
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