A one-dimensional harmonic oscillator of mass M and spring constant k is subjected to a friction force F_f (t) = −λv (t) where v (t) is the velocity of the point mass and λ > 0. Using a coordinate axis Ox where the origin O corresponds to the position of the point mass when the harmonic oscillator is at rest, the equation of motion reads,

a) Express the mechanical energy E (t) in terms of the coefficients k, C and γ.

b) Calculate the power P (t) dissipated due to the friction force F_f (t) during one oscillation period.

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x + 2yx+wzx = 0 where wf = k/M and y = X/ (2M). In the weak damping regime, where y << wo, the position can be expressed as x (t) = Cecos (wot + p) where C and are integration constants.