# Write a note on conservation of angular momentum?

Nirmal Singh.
10 years ago

Thelaw of conservation of angular momentumstates that when no externaltorqueacts on an object or a closed system of objects, no change of angular momentum can occur. Hence, the angular momentum before an event involving only internal torques or no torques is equal to the angular momentum after the event. This conservation law mathematically follows fromisotropy, or continuous directional symmetry of space (no direction in space is any different from any other direction). SeeNoether's theorem.[4]

The time derivative of angular momentum is calledtorque:

$\mathbf{\tau} = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \times \mathbf{p} + \mathbf{r} \times \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = 0 + \mathbf{r} \times \mathbf{F} = \mathbf{r} \times \mathbf{F}$

(The cross-product of velocity and momentum is zero, because these vectors are parallel.) So requiring the system to be "closed" here is mathematically equivalent to zero external torque acting on the system:

$\mathbf{L}_{\mathrm{system}} = \mathrm{constant} \leftrightarrow \sum \mathbf{\tau}_{\mathrm{ext}} = 0$

where$\mathbf{\tau}_{ext}$is any torque applied to the system of particles. It is assumed that internal interaction forces obeyNewton's third law of motionin its strong form, that is, that the forces between particles are equal and opposite and act along the line between the particles.

In orbits, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit:

$\mathbf{L}_{\mathrm{total}} = \mathbf{L}_{\mathrm{spin}} + \mathbf{L}_{\mathrm{orbit}}$;

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

Regards,

Nirmal Singh