7

Moment of Inertia and Mutilation Studies of an Insect Wing

99

Appendix

This part of the appendix is extracted and modified from “Properties of Matter”

written by P. E. SubramaniaIyer [3]

Energy of a Particle Executing Simple Harmonic Motion

d²x

dt^{2 = −ω2x (Linear Motion)}

If m—the mass of the particle and x—displacement,

the force necessary to produce this acceleration is m ^d2x

dt^{2 .}

If it undergoes a small additional displacement dx, work done by the force for

producing the additional displacement is given by

dw = F.dx = ω²m.x.dx

Assuming that the whole of the displacement is produced this way, the work done

is

∫dw =

x

∫

0

ω²m.x.dx = ¹

2^ω2mx2

This work gives the potential energy of the particle at that instant. The

instantaneous velocity, v, of the particle is given by

v = ^dx

dt ^{and P.E = 1}

2^ω2mx2

v² = ω^2

a² −x^2

where a—amplitude of the Simple Harmonic Motion (S.H.M) and hence

v = ω



a² −x²



(7.6)

The kinetic energy of the particle at that instant is equal to ¹

2^mv2

K.E = ¹

2^mω2

a² −x^2

The total energy of the particle at the instant is