Question

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?

Answer 1

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.