Question

In which of the following cases, the angular momentum is conserved?
A. The Planet Neptune moves in an elliptical orbit around the sun with the sun at one focus.
B. A solid sphere rolling on an inclined plane.
C. An electron revolving around the nucleus in an elliptical orbit.
D. An $\alpha$ particle approaching a heavy nucleus from sufficient distance.

Answer 1

In order to solve this question, we have to understand definition of angular momentum which states that angular momentum is a measure of rotation of body and it is defined as product of moment of inertia and body’s angular velocity around the axis which is rotating. Moment of inertia is defined as the body's resisting capacity against angular acceleration.

Complete step by step answer:
For a conservation of angular momentum we would define torque. Torque is defined as force with which the body is rotating and it is defined as change in angular momentum per unit time.
$\vec \tau = \dfrac{{d\vec J}}{{dt}}$
So if $\vec \tau = 0$
$\Rightarrow \dfrac{{d\vec J}}{{dt}} = 0$
$\Rightarrow \vec J =$ constant
So we have to see which option $\vec \tau = 0$.

For option [A], since Neptune rotates around sun in elliptical orbit so line of focus passes through sun and hence torque about sun is zero so angular momentum is conserved.

For option [B], since the sphere is rolling about its Center Of Mass but the force of friction at contact point is non zero so torque due to friction is non zero and hence angular momentum not conserved.

For option [c], the electron revolves around the nucleus due to coulombic attraction which is due to intrinsic charge of the electron so no external force hence torque is zero so angular momentum is conserved.

For option [D], since the alpha particle is approaching by translating not rotating so torque is zero and hence angular momentum of the body is conserved.

So correct options are A, C and D.

Note: It should be remembered that torque is defined as the cross product of force and the perpendicular distance of the axis from the line of action of force so either of them is zero net torque is zero so angular momentum is conserved. Also if rotating bodies translate then torque is not $0$ in this case angular momentum is not conserved.