Angular Momentum Formula – Solved Example

In this post on the angular momentum formula, some solved examples are provided for a better understanding of the concepts of the angular momentum formula and how to use it to solve the given problems. Here in the Angular Momentum Formula post, The solved examples provided involve torque, rotational momentum, rigid body motion, classical mechanics, and mechanics. Let us try to solve a few problems for this topic of angular momentum.
When we imagine that gravitational force is not a central force, then what happens to the calculations of angular momentum is explained here. This condition is put in the problem created by me to understand the various concepts involved in understanding the angular momentum topic.

Q. Angular Momentum Formula when Gravitational force is not central force.

Assuming that gravitational force is not central force and is under the influence of a tangential swirl component. Potential energy is given as $U(r,\mathit{\theta })=-\frac{\mathrm{GMm}}{r}(1+\mathit{\alpha }\mathit{\theta })$. The distance between center of planet and Earth center is (3r+7R) where r= radius of planet and R= Radius of earth. Find the final angular momentum if the initial angular momentum is zero and the radial velocity at grazing is zero.

The problem and the final solution is put in this compact image as below.

The above are imaginary positions when gravitational force is not a central force. Let us try to solve this problem and understand the science behind it. Hereby I encourage others to do the required research work and comment if the gravitational force deviates from central force principles.
I hereby give the solution to the best of my knowledge. Changes and suggestions are welcome.
We dwell on the problem and solve it.

Solution for final angular momentum formula

From the given data for finding the final angular momentum formula we get,

Initial separation = $r_i=3r+7R$

Initial angular momentum = $L_i=0$

Initial radial velocity = ${\dot{r}}_i=0$

Grazing separation, $r_f=r+R$

Grazing radial velocity = ${\dot{r}}_f=0$

$U(r,\mathit{\theta })=-\frac{\mathrm{GMm}}{r}(1+\mathit{\alpha }\mathit{\theta })$

${\mathit{\theta }}_i=0$

Reduced mass of the system = $\mathit{\mu }=\frac{\mathrm{Mm}}{(M+m)}$

The general equation for total energy is given by,

$E=K+U=\frac{1}{2}\mathit{\mu }{\dot{r}}^2+\frac{L^2}{2\mathit{\mu }{r}^2}+U(r,\mathit{\theta })$

By using the suitable subscripts i for initial state and f for final state, we get,

Initial state Energy Equation for calculations using angular momentum formula

From above given conditions of initial state,

$E_i=0+0-\frac{\mathrm{GMm}}{r_i}(1+\mathit{\alpha }(0\mathrm{))}=-\frac{\mathrm{GMm}}{r_i}$

Final state Energy Equation for calculations using angular momentum formula

$E_f=\frac{1}{2}\mathit{\mu }(0{)}^2+\frac{L_f{}^2}{2\mathit{\mu }r_f{}^2}-\frac{\mathrm{GMm}}{r_f}(1+{\mathit{\alpha }\mathit{\theta }}_f)=\frac{L_f{}^2}{2\mathit{\mu }r_f{}^2}-\frac{\mathrm{GMm}}{r_f}(1+{\mathit{\alpha }\mathit{\theta }}_f)$

Conservation of Energy principle

By applying the conservation of energy principle, $E_i=E_f$ , we get,

$-\frac{\mathrm{GMm}}{r_i}$ = $\frac{L_f{}^2}{2\mathit{\mu }r_f{}^2}-\frac{\mathrm{GMm}}{r_f}(1+{\mathit{\alpha }\mathit{\theta }}_f)$

From above , we get,

$L_f{}^2=2\mathit{\mu }{\mathrm{GMmr}}_f[1+{\mathit{\alpha }\mathit{\theta }}_f-\frac{r_f}{r_i}]$

But $r_i=3r+7R$ and $r_f=r+R$, by substituting these values in above equation , we get,

$L_f{}^2=2\mathit{\mu }\mathrm{GMm}(r+R\mathrm{)[}\frac{2(r+3R)}{(3r+7R)}+{\mathit{\alpha }\mathit{\theta }}_f]$

From above the final Angular Momentum is given as above.

The above is the answer for the final angular momentum formula.

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