Lost and in Need of Ideas for Finding Inspiration

[fer Mnaj]

Homework Statement

If we have motion in a circular orbit of a body of mass m subject to the gravitational force of the Sun.
Consider the approximation for M >> m in which the Sun is located at the center of mass and μ = m.



How to show that the total mechanical energy associated with the movement of this body fulfills that
$E = −1 / 2U$ ?

where U is the gravitational potential energy of m due to M

Relevant Equations

$-∇U= -G Mm/r^2$

Any idea? I am pretty lost

 

[PeroK]

Are you able to calculate the KE of the mass in a circular orbit?

 

[fer Mnaj]

Not really, can you explain it?

 

[PeroK]

Not really, can you explain it?

What is needed to keep an object in a circular orbit?

 

[jbriggs444]

Not really, can you explain it?

If you know that the orbit is circular, you have some formulas that could be used to express the required centripetal force.

 

[fer Mnaj]

$m (v^2/r)$

 

[etotheipi]

By $E$, I suspect you mean kinetic energy instead of total mechanical energy.

 

[fer Mnaj]

why?

 

[etotheipi]

Because if $E = U + T$, then for circular motion the following holds at all times: $E = \frac{1}{2}U = -T$.