Answer to Question #164541 in Physics for Hridita Chowdhury

Question

1.A particle showing damped harmonic motion is subjected to an external periodic force.Establish the differential equation of motion of the particle.Explain each terms & solve it to obtain an expression for the displacement of the oscillation.(Diagrams are necessary)

2.Explain the phenomena of quality factor & sharpness of resonance of forced harmonic oscillator & clearly explain the factor on which the sharpness of resonance depends.(Diagrams are necessary)

Answer 1

1)

$$F _ d ​ =−bv$$

$$F =ma =−kx−mbv$$

$$\omega^2=\frac{k}{m}-\frac{b^2}{4m^2}$$

$$x=Ae^{-\frac{bt}{2m}}\cos{(\omega t+\phi_0)}$$

2) The sharpness of resonance can be understood better by understanding resonance. Resonance is defined as the tendency of a system to vibrate with an increase in amplitude at the excitation of frequencies. Resonance frequency or resonant frequency is the maximum frequency at which the amplitude is relatively maximum. The Q factor is used to define the sharpness of the resonance.

The sharpness of resonance is defined using the Q factor which explains how fast energy decay in an oscillating system. The sharpness of resonance depends upon:

Damping: Effect due to which there is a reduction in amplitude of vibrations

Amplitude: Maximum displacement of a point on a vibrating body which is measured from its equilibrium position.

The sharpness of resonance increases or decreases with an increase or decrease in damping and as the amplitude increases, the sharpness of resonance decreases.

Q factor or quality factor is a dimensionless parameter that is used to describe the underdamped resonator and characterizes the bandwidth and center frequency of the resonator.