Devise a variational state function for a particle with mass m executing simple harmonic motion and hence evaluate its energy???
I will be explaining you with the help of potential energy
Potential energy is the energy possessed by the particle when it is at rest. Let’s learn how to calculate the potential energy of a particle performing S.H.M. Consider a particle of mass mperforming simple harmonic motion at a distance x from its mean position. You know the restoring force acting on the particle is F= -kx where k is the force constant.
Now, the particle is given further infinitesimal displacement dx against the restoring force F. Let the work done to displace the particle be dw. Therefore, The work done dw during the displacement is
dw = – fdx = – (- kx)dx = kxdx
Therefore, the total work done to displace the particle now from 0 to x is
∫dw= ∫kxdx = k ∫x dx
Hence Total work done = 1/2 K x2 = 1/2 m ω2x2
The total work done here is stored in the form of potential energy.
Therefore Potential energy = 1/2 kx2 = 1/2 m ω2x2.
I hope this helpsss!!
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