100

N. Chari et al.

= ¹

2^{ω2mx2 + 1}

2^ω2m



a² −x^2

Total Energy = ¹

2^ω2ma2

(7.7)

The ω = ^2π

T ^{and ω2 =}

 2π

T

2 = 4π2

T ^{2 , where T = time period or time taken for one}

cycle.

The total energy of the particle is given by

E = ^2π2ma2

T²

(7.8)

Thus, the total energy E at any instant is constant and independent of local

displacement x for small amplitudes.

If ϑ be the frequency of the particle, ϑ = ¹

T ⁼

ω

2π

∴Total Energy of particle (E) = 2π²ma²ϑ²

(7.9)

Moment of Inertia

The Moment of Inertia of a body depends on

(a)

Mass of the body and

(b)

The distance of the mass from the axis of rotation

I = mass × (radius of gyration)²

(7.10)

The radius of gyration, k, is the distance of the centre of mass from the fulcrum.

The dimensional formula for the moment of inertia is I = Mk².

Comparison between Translatory and Rotary Motions is presented in Table 7.3 at

the end.