Non-equal Gravitational Potential and Kinetic Energy in a closed system

[FlipC]

While trying to get my head around Gravitational Potential Energy I devised the following simple system:

Point Mass A of 1kg is 1000m away from Point Mass B of 100kg within an empty universe.

The gravitational force exerted by A on B is G*10^-16; by B on A is G*10^-4. At time=0 these two unmoving bodies possesses no kinetic energy. Both posses the same amount of gravitational potential energy (mgh) with respect to each other of G*10^-1.

At t=1 second they've attracted each other and moved closer. Body A at a speed of G*10^-4 and B at a speed of G*10^-8. They therefore posses kinetic energy equal to .5mv^2. Their potential energy has also changed due to both the reduced distance and the increase in relative gravities.

However when I run the figures I end up with ~2.2*10^-29 Joules over.

Given the tiny value it may be rounding errors in my calculation programme or a flaw in my basic assumptions, but if not where has this energy come from?

 

[Doc Al]

The gravitational force exerted by A on B is G*10^-16; by B on A is G*10^-4.

How did you calculate this? They must exert the same force on each other. (Newton's 3rd law.)

At time=0 these two unmoving bodies possesses no kinetic energy. Both posses the same amount of gravitational potential energy (mgh) with respect to each other of G*10^-1.

The gravitational PE belongs to the system of both bodies, not to each individually. (And you cannot use 'mgh' to calculate it, of course.)

At t=1 second they've attracted each other and moved closer. Body A at a speed of G*10^-4 and B at a speed of G*10^-8.

Since momentum is conserved, their speeds will always be such to give zero total momentum.

 

[FlipC]

How did you calculate this? They must exert the same force on each other. (Newton's 3rd law.)

Tcch yeah just realized that, force = G*10^-4 with speeds of G*10^-4 and G*10^-6 respectively.

The gravitational PE belongs to the system of both bodies, not to each individually. (And you cannot use 'mgh' to calculate it, of course.)

PE with relation to what and how is it then calculated?

 

[Doc Al]

PE with relation to what and how is it then calculated?

The gravitational PE of the system is given by:
$$U = - \frac{Gm_1m_2}{r}$$

 

[FlipC]

Okay having calculated that at t=0 and t=1 determine KE at both times as well. I still get a discrepancy. There's more KE than be accounted for by the change in PE by 2.2*10^-29J

 

[Doc Al]

Okay having calculated that at t=0 and t=1 determine KE at both times as well. I still get a discrepancy. There's more KE than be accounted for by the change in PE by 2.2*10^-29J

How are you doing your calculation? How far have the bodies moved? What's the change in PE? (Is 2.2*10^-29J significant or just a trivial difference due to round off?)

 

[FlipC]

After 1 second Body A will be traveling at G*10^-4 m/s and Body B at G*10^-6 m/s. Assuming linear acceleration the distance between them will have reduced by (G*10^-4+G*10^-6)/2 m.

Calculate PE and KE at t=0 and t=1. Given a closed system the difference in sum of energies at each point should be zero; except I keep getting that difference in total KE being higher than total PE.

Hmm okay writing that out again. Gravity changes with distance therefore the acceleration changes such that the bodies move faster and the distance increases more. However it still seems that the ratios between them produce a higher KE than allowed for by the change in PE.

As I said it may well be a rounding error.