In simple harmonic motion

PHYSICS · Multi-part · A Level · Physics · Measurements and their errors · Measurements and their errors

Problem

In simple harmonic motion, the displacement of a particle from the mean position at time

t

is written as

x = A \sin(\omega t + \phi)

, where

A

is amplitude,

\omega

the angular frequency and

\phi

the phase constant. (i) Show that the quantity

(\omega t + \phi)

must be dimensionless. (ii) Hence obtain the dimensions of

\omega

and comment on its relation with the time period of oscillation. (iii) A simple pendulum of length

\ell

has time period

T = 2\pi \sqrt{\dfrac{\ell}{g}}

. Using your result from part (i), verify that the expression

2\pi\,\dfrac{t}{T}

appearing in the argument of the sine function for a pendulum is dimensionless.