Angular Momentum – Definition, Formula and Unit
The turning movement of a particle about the axis of rotation is called the angular momentum of the particle. It is measured as the product of the linear momentum and perpendicular distance of its line of action from the axis of rotation.
It is denoted by $\displaystyle L$.
Angular Momentum Formula
The formula of angular momentum is
angular momentum = linear momentum × perpendicular distance from the axis of rotation
i.e. $\bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol {\displaystyle L=r\times p=m\left( {r\times v} \right)}}$
where r is the position vector of the particle with respect to O. As the particle moves relative to O, in the direction of its linear momentum p (= mv), position vector r rotates around O.
The formula in terms of moment of inertia is
angular momentum = moment of inertia × angular velocity
i.e. $\bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol {\displaystyle L=I\omega}} $
where I is the moment of inertia and $ \displaystyle \omega $ is the angular velocity.
Unit of Angular Momentum
The S.I. unit of angular momentum is kg m2 s-1
Its dimensional formula is [M L2 T-1]
Angular momentum is a vector quantity. The direction of the angular momentum vector is always perpendicular to the plane formed by the vectors r and p.
Law of Conservation of Angular Momentum
It states that if no external torque acts on a system, then the total angular momentum of the system always remains conserved.
Consider a systme of n-particles. Suppose that the particles of the system are under the action of torques due to external forces acting on them. If $\displaystyle \frac{d}{{dt}}\left[ {\overrightarrow{L}(tot)} \right]$ is the rate of change of angular momentum of the system, then the total torque acting on the system is given by
$\displaystyle \overrightarrow{\tau }(tot)=\frac{d}{{dt}}\left[ {\overrightarrow{L}(tot)} \right]$
If, the total torque due to external forces on the system is zero, then
$\displaystyle \frac{d}{{dt}}\left[ {\overrightarrow{L}(tot)} \right]=0$
or
$\displaystyle \overrightarrow{L}(tot)$ = a constant vector
If $ \displaystyle {{\overrightarrow{L}}_{1}}$, $\displaystyle {{\overrightarrow{L}}_{2}}$, $\displaystyle {{\overrightarrow{L}}_{3}}$,…$\displaystyle {{\overrightarrow{L}}_{n}}$ are the angular momenta of the different particles of the system about the axis of rotation, then
$ \displaystyle {{{\vec{L}}}_{1}}+{{{\vec{L}}}_{2}}+{{{\vec{L}}}_{3}}+\text{ }\ldots \text{ }+{{{\vec{L}}}_{n}}$ = a constant vector
Since angular momentum is a vector quantity; in the absence of external torque, both its magnitude and direction must remain unchanged.
For example, when a planet moves around the sun, the torque on the planet is due to the sun, i.e., from within the system. As no external torque is acting on the planet, the constant magnitude of the angular momentum vector leads to the constant areal velocity of the planet (i.e., Kepler’s second law of planetary motion) and the fixed direction of the angular momentum vector points out that the plane of the orbital motion of the planet around the sun also must remain fixed.
It may be pointed out that if external forces do not act on the system, then both linear momentum and angular momentum of the system remain constant.
Examples of Law of Conservation of Angular Momentum
A person carrying heavy weights in his hands and standing on a rotating platform can change the speed of the platform. The figure below shows the weights in arms which are outstretched. When the person suddenly folds his arms, his moment of inertia ($I$) decreases. As angular momentum ($I \times \omega$) remains constant, therefore, angular velocity ($\omega$) of his body increases.
Suppose a ballet dancer is rotating with her arms and legs stretched outwards. When she folds her arms and brings the stretched leg close to the other leg, her moment of inertia decreases and hence her angular velocity increases.
