Why is Kinetic Energy = (-) Total Energy and Potential Energy = 2 $\times$ Total Energy?

Question

I came across this relation while reading on the Bohr atomic model. Are there any other forces for which these relations hold good?

Answer 1

You've discovered the virial theorem.

The virial theorem tells us that for a bound system where the potential energy $V$ is given by an equation:

$$ V(r) \propto r^{-n} $$

The average kinetic energy $T$ and average potential energy $U$ are related by:

$$ 2T = -nU $$

For the electrostatic force $V(r) \propto r^{-1}$ so $n = 1$ and:

$$ 2T = -U \tag{1} $$

The total energy is $E = T + U$ so using equation (1) to substitute for $U$ gives us:

$$ E = -T$$

and substituting for $U$ gives:

$$ E = \tfrac{1}{2}U $$

The example you've found is for the electrostatic force, but exactly the same applies to the gravitational force. Indeed, the first evidence for dark matter was when Fritz Zwicky used the virial theorem to show that the velocities in a galaxy cluster were higher than they should be.

Answer 2

This is true for central potential problems. Because in that case, both potential energy and kinetic energy are in (1/r) terms. Also, Potential energy is negative and its magnitude is twice that of Kinetic energy . Thus we conclude that only the mathematics of this problem allows it to be expressed as the formula u proposed and it is general for any central force problems