Average Kinetic & Potential Energy in Simple Harmonic Motion

[sadhu]

in SHM
average K.E when done w.r.t to time is equal to average potential energy calculated w.r.t to time.

but today in class when my sir asked me to to prove average K.e = average P.E
I just tried integrating it w.r.t to displacement and then divided it with A ,
I thought this gives us av . energies in 1/4 vibration and as all the four parts are identical
average will remain same , but i found that av . K.E=2*av. P.E

I then done this again w.r.t to time and got the answer

but still I am confused , about how can that come i.e K=2*P.E

 

[Greg Bernhardt]

In simple harmonic motion, the potential energy of an object is given by PE = (1/2)kx^2. Because there is a relationship between displacement and velocity (x = A cos(wt)), you can use calculus to calculate the average kinetic energy of an object in SHM by taking the time derivative of the potential energy equation.

The average kinetic energy of an object in SHM is equal to twice the average potential energy. This is because the kinetic energy of an object is proportional to its velocity squared, while the potential energy is proportional to its displacement squared. Since the velocity of an object in SHM is changing twice as fast as its displacement, the average kinetic energy must be twice the average potential energy.