Vibrating Particle Total Energy

∴ The kinetic energy of the particle at the instant the displacement is y.

= ½ mv²

= ½ m.ω² (a² – y²)

Potential energy of the vibrating particle is the amount of work done in overcoming the force through a distance y.

Acceleration           = – ω²y

Force                       = – mω²y

(The –ve sign shows that the direction of the acceleration and force are opposite to the direction of motion of the vibrating particle).

Total energy of the particle at the instant the displacement is y

= K.E. + P.E.

= ½ mω² (a² – y²) + ½ mω²y²

= ½ mω²a²

= ½ m(2πn)2a²

= 2π²ma²n²

As the average kinetic energy of the vibrating particle = π²ma²n², the average potential energy = π²ma²n². The total energy at any instant is constant.

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