Question

To find the expression of kinetic energy potential energy and Total energy of a system excluding SHM​

Answer 1

Answer:

ne can't discuss the difference between factors of 1/3 and 1/6 without mathematics. The difference between these two numbers – and generally, any fact about any numbers – is all about mathematics.

If all values of x between 0 and A were equally likely, the average value of kx2/2 would be kA2/6 as you say because the average value of X2 for X uniformly distributed between zero and one is

∫10dXX2=X33∣∣∣10=1/3

However, when the harmonic oscillator (you talk about a spring which is a harmonic oscillator) oscillates, it oscillates harmonically, via sines and cosines, so it spends much more time near the |x|=A extreme points where the speed is low than it spends in the vicinity of x=0 where the speed is high.

If you compute the average value (over time) of kx2/2 in this oscillating motion, the result will be proportional to the average value of cos2t over time which is equal to 1/2. So the average kinetic energy of the oscillating motion will be kA2/4 – a number that doesn't appear in your list of results at all.

Similarly, the maximum value of the kinetic energy is mv2max/2=kA2/2 at the maximum achieved exactly when the potential energy kx2/2 has the minimum value (zero). Similarly, the minimum value of the kinetic energy is 0 exactly when the potential energy kx2/2 is maximized i.e. at |x|=A.

The average contribution of the kinetic and potential energy to the total energy is the same for the harmonic oscillator – both averages are kA2/4 – a fact that is guaranteed by the "virial theorem".

No factor of 1/3 ever appears in the correct average values for the harmonic oscillator (just like it doesn't appear in the right solution to the sleeping beauty problem).