Question

The energy E of a particle oscillating in S.H.M depends on the mass m of particle ,frequency n and amplitude a of oscillation .show dimensionally that E proportional ti m.nsquare .a square .

Answer 1

Hi..
The total energy (E) of an oscillating particle is equal to the sum of its kinetic energy and potential energy if conservative force acts on it.

The velocity of a particle executing SHM at a position where its displacement is y from its mean position is v = ω√a2 – y2

Kinetic energy
Kinetic energy of the particle of mass m is,

K = ½ m [ω√a2 – y2]2......(1)..


Potential energy
From definition of SHM F = –ky the work done by the force during the small displacement dy is dW = −F.dy = −(−ky) dy = ky dy

∴ Total work done for the displacement y is,

W = \int dW = \int_{0}^{y}ky dy


As k = mω2, therefore,

W =\int_{0}^{y}m\omega ^{2}y dy

Thus, W = ½ mω2y2 …... (2)

This work done is stored in the body as potential energy.

…... (3)..
Total energy, E = K+U

= ½ mω2 (a2 – y2) + ½ mω2y2

= ½ mω2a2

So,

…... (4)

Thus we find that the total energy of a particle executing simple harmonic motion is ½ mω2a2...

Hope this helps u!!