Subject:- Mechanical Vibration

Topic:- Basic Concepts of Vibration

Difficulty:- Medium

Total energy of the given system -

T = Kinetic Energy due to translatory motion + Kinetic Energy due to rotatory motion + Potential Energy of spring

$T = \frac{1}{2}mx^2 + \frac{1}{2}Iθ^2 + \frac{1}{2}kx^2 $

$T = \frac{1}{2}mr^2θ^2 + \frac{1}{2}.\frac{1}{2}mr^2θ^2 + \frac{1}{2}kr^2θ^2 $

$T = \frac{1}{2}mr^2θ^2 + \frac{1}{4}mr^2θ^2 + \frac{1}{2}kr^2θ^2 $

$T = \frac{3}{4}mr^2θ^2 + \frac{1}{2}kr^2θ^2 = constant $

Differentiating T with respect to time we get,

$0 = \frac{3}{4} . 2mr^2θ + kr^2θ = 0$

$ i.e. = \frac{3}{2}mr^2θ + kr^2θ = 0$

$ Ω(n) = \sqrt{\frac{kr^2}{3/2mr^2}}$

$ i.e. = \sqrt{\frac{2k}{3m}} rad/sec$

$ f(n) = \frac{1}{2π}\sqrt{\frac{2*4000}{3*4m}} = 4.11 Hertz(Hz)$