Physics

Problem

The displacement of a damped oscillator as a function of time is given by

x = A e^{-\left(\dfrac{b}{m}\right) t} \cos(\omega t)

, where

A

is the initial amplitude,

b

is a damping constant,

m

is the mass of the oscillator and

\omega

is the angular frequency. (i) Determine the dimensions of the combination

\dfrac{b}{m}

. (ii) Show that the factor

e^{-\left(\dfrac{b}{m}\right) t}

is dimensionless. (iii) Verify that the entire equation is dimensionally consistent. a. (i) Determine dimensions of

\dfrac{b}{m}

. b. (ii) Show

e^{-\left(\dfrac{b}{m}\right) t}

is dimensionless. c. (iii) Verify dimensional consistency of the equation.