Is the Law of Areas Still Applicable with a Changed Law of Gravitation?

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

I think if the law of gravitation changes then the law of areas should still hold .Law of areas is nothing but law of conservation of angular momentum . Since the changed law of gravitation is still central the law of areas should still hold true .

But the law of period will change as the dependency on the distance changes .

So option 1) is right .

Is that correct ?

 

[Charles Link]

I only looked at it somewhat quickly, but I believe it is correct. The law of areas is derived from conservation of angular momentum. Writing a circular orbit with $\frac{AMm}{r^3}=\frac{mv^2}{r}$ with $v=\frac{2 \pi r}{T}$ would give you $\frac{r^4}{T^2}=constant$.$\\$ An additional google of the topic seems to suggest that $n=3$ would normally result in an unstable orbit. Further study is needed... See https://www.reddit.com/r/askscience/comments/74okkx/what_would_orbital_mechanics_be_like_if_gravity/ $\\$ I think in any case (1) may still be correct.

 

[Jahnavi]

Thank you .

Is potential energy defined in the changed law just like the way it is defined usually (-GMm/r) ?

I mean , does potential energy exist if the force has an inverse cubic relation ?

 

[Charles Link]

Thank you .

Is potential energy defined in the changed law just like the way it is defined usually (-GMm/r) ?

I mean , is potential energy defined if the force has an inverse cubic relation ?

It would be $U(r)= -\int\limits_{r}^{+\infty} \frac{AMm}{r^3} \, dr=-\frac{AMm}{2 r^2}$

 

[Jahnavi]

OK . So potential energy has nothing to do with inverse square law . Right ?

 

[Charles Link]

OK . So potential energy has nothing to do with inverse square law . Right ?

It could still be a conservative force, where $\vec{F}=-\nabla U$. That does not require $\vec{F}$ to be inverse square. Inverse cube works ok for that as well, i.e. the function $U$ that is computed above will act as required for potential energy. The important thing here that it is a "conservative" potential in that the sum of the kinetic and potential energy is conserved.

 

[Jahnavi]

OK .

Suppose instead of force having inverse square/cube dependancy , potential energy is U = -K₁r³ and the attractive central force is F = K₂r² , is the potential energy defined and does conservation of energy still hold true ?

Sorry , if my questions look irrelevant but we do get such type of questions as is evident from the OP . I am just trying to understand this better .

 

[Charles Link]

If $3 K_1=K_2$, I believe that would work. (And it's a good question). $\\$Edit: But in that case, one correction: $U=+K_1 r^3$. (Just like for a spring: $U=+\frac{1}{2} k r^2$, with a "+" sign, when the attractive force is $F=kr$.).

 

[Jahnavi]

Does conservation of energy also hold for the case in post#7 ?

Does that mean any central force of the form Krⁿ , where n is a non zero integer will be conservative if it satisfies F = -dU/dr ?

 

[Charles Link]

Does conservation of energy also hold for the case in post#7 ?

Does that mean any central force of the form Krⁿ , where n is a non zero integer will be conservative if it satisfies F = -dU/dr ?

I believe that is the case. In general, if the force comes from the (minus) gradient of a potential function, it will be conservative.

 

[Jahnavi]

Thanks !