Question

State the law of conservation of angular momentum.

Answer 1

Hint: The law of conservation of angular momentum talks about how the total angular momentum of a system is conserved when no net external torque acts on it. It is the rotational analogue of the law of conservation of linear momentum and can be defined in the same way, just by changing all the variables into their rotational analogues (for example changing force to torque).

Formula used:
According to the law of conservation of angular momentum, when no net external torque acts on an object,
$L=\text{constant}$
$\therefore \Delta L=0$
where $L$ is the total angular momentum of the system.
The net external instantaneous torque $\tau$ acting on a system is given by,
$\tau =\dfrac{dL}{dt}$
where $t$ is the time period for which the torque is applied on the object and $L$ is its angular momentum. Therefore, torque is the rate of change of angular momentum.

Complete step by step answer:
The law of conservation of angular momentum states that, when the net external torque acting on a system is zero, its total angular momentum is conserved and hence, does not change.
Therefore, according to the law of conservation of angular momentum, when no net external torque acts on an object,
$L=\text{constant}$ --(1)
$\therefore \Delta L=0$
where $L$ is the total angular momentum of the system and $\Delta L$ is the change in it.
This can be proved from the mathematical definition of torque. , torque is the rate of change of angular momentum.
The net external instantaneous torque $\tau$ acting on a system is given by,
$\tau =\dfrac{dL}{dt}$ --(2)
where $t$ is the time period for which the torque is applied on the object and $L$ is its angular momentum.
Now, if the net external torque is zero, $\tau =0$, from (2), we get,
$\dfrac{dL}{dt}=0$
$\therefore dL=0$
$\therefore L=\text{constant}$
Comparing with (1), we see that the law of conservation of angular momentum has been proved.

Note: The law of conservation of angular momentum can come in handy while solving many various rotational mechanics problems. By assuming a proper system, where the only torques acting, are internal torques, many unnecessary variables can be done away with it. By using the law of conservation of angular momentum, many equations can be devised to get relations between the different variables.
Before proceeding to apply the law of conservation of angular momentum, students must properly analyze whether there are any net external torques or not. In the presence of even a single torque, the law is not valid since the angular moment of the system is bound to change.