Question

Derive, from the concept of the total energy of a system, the total energy of an oscillating system in terms of k and amplitude. (show all steps begging to end on how you get the equation and answer)

Answer 1

Let us consider a Simple pendulum in SHM , oscillating with amplitude of A from its mean position on either side of the mean position

we know that the simple harmonic oscillation , satisfies the condition that the restoring force and the displacement as

   F proportional to the displacement and directed in the opposited direction ( that is towards the mean position always)

     

   F = -k*x

say the simple pendulum oscillating with a velocity of v at any point between the extream ends (x displacement from mean position) the velocity is given by

   v = w (A^2-x^2)

squaring on both sides

   v^2 = w^2 (A^2-x^2)
we know that the kinetic energy is k.e = 0.5*m*v^2

   k.e = 0.5*m* w^2 (A^2-x^2)

from the relation between the k,m,and w , w = sqrt(k/m)

   k.e = 0.5*k(A^2-x^2) ---------------(1)

now for the potential energy

   here the work done on the oscillator to make a displacement of dx from mean position and the total work done to reach the extream end we get it by integrating , taht is

   dW = -F*dx

   W = integral(-(-kx)dx)

   W = k*x^2/2

   W = 0.5*k*x^2 ----------------------(2)

now the total energy of the system at any point is sum of kinetic and potential energy

   E = k.e+p.e

   E = (1)+(2)

   E = 0.5*k(A^2-x^2) +0.5*k*x^2

   E = 0.5*k*A^2

this is the energy of the oscillator under SHM , interms of k and A