Show that the total energy of a particle performing linear SHM is dire

1.	Show that the total energy of a particle performing linear SHM is directly proportional to (1) the square of the amplitude(2) the square of the frequency.
Answer» For a particle of mass m executing SHM with angular frequency ω and amplitude A, its kinetic and potential energies are respectively, KE = 1/2 mω² (A² – x²) … (1) and PE = 1/2 mω²x² … (2) Then, the total energy, E = PE + KE = 1/2 mω² x² + 1/2 mω² (A² – x²) = 1/2 mω² A² …. (3) Therefore, total energy of the particle is 1. directly proportional to the mass (E ∝ m), 2. directly proportional to the square of the amplitude (E ∝ A²) 3. proportional to the square of the frequency (E ∝f² ), as f = ω/2π

Show that the total energy of a particle performing linear SHM is directly proportional to (1) the square of the amplitude(2) the square of the frequency.

Answer»

For a particle of mass m executing SHM with angular frequency ω and amplitude A, its kinetic and potential energies are respectively,

KE = 1/2 mω² (A² – x²) … (1)

and PE = 1/2 mω²x² … (2)

Then, the total energy,

E = PE + KE

= 1/2 mω² x² + 1/2 mω² (A² – x²)

= 1/2 mω² A² …. (3)

Therefore, total energy of the particle is

1. directly proportional to the mass (E ∝ m),

2. directly proportional to the square of the amplitude (E ∝ A²)

3. proportional to the square of the frequency (E ∝f² ), as f = ω/2π

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