Question

State and prove the principle of conservation of angular momentum explain the conservation of angular momentum with examples.

Answer 1

Answer:

Principle Of Conservation Of Angular Momentum:

It states that in absence of any external torque on the system , the angular momentum of that system remains constant.

$\boxed{ \sf{ \blue{ \large{ \bold{Mathematical \: Proof}}}}}$

Let's consider a rotating object on whose the external torque is zero.

Torque is the rotational analogue of acceleration. Let angular momentum be denoted as L and time by t.

$\sf{ \red{ \therefore \: \tau \: = 0}}$

$\sf{\implies \: \dfrac{dL}{dt} = 0}$

$\sf{ \implies \: dL = 0 \times dt}$

$\sf{ \implies \: \int dL = \int 0 \times dt}$

$\sf{ \implies \: L2 - L1 = 0}$

$\sf{ \implies \: L2 = L1 = constant}$

So angular momentum remains constant.

$\boxed{ \orange{ \sf{ \bold{ \large{Examples : }}}}}$

1. When a ball dancer rotates on her toes, she doesn't have external torque acting on her. Hence her Angular Momentum remains constant .

Hence, when she decreases her Moment of Inertia , her Angular Velocity increases and she rotates faster.

2. Similar things happen with swimmers and divers.

Answer 2

Answer:

Principle:

Consider any rotating object. If there is no external torque acting on the object , it willhave constant angular momentum.

If Torque = 0

=> dL/dt = 0

=> L = constant

So angular momentum remains constant after integration of the function.

Examples :

While somer-saulting, the person decreases the moment of Inertia by pulling hands and legs towards himself/herself. This increases Angular velocity and hence completes the somer-sault.