Question

x = A sin (w t + c), where x is displacement and t is time. What are the units and dimensions of A, w and c?
(w is omega)​

Answer 1

Given :

Displacement equation of a particle is given by

• x = A sin (ωt + c)

To Find :

The units and dimensions of A, ω and c.

Solution :

❖ The principle of homogeneity of dimensions is used in checking the correctness of formulae, and also in the derivation of formulae.

• Only like quantities having the same dimensions can be added to ot substracted from each other.

♦ Functions like sin, cos are dimensionless.

$\sf:\implies\:[x] = [A]\cdot[1]$

$\bf:\implies\:[A]=[L^1]$

• Unit of A = m

➙ Dimension of ω = [T‾¹]

➙ Unit of ω = rad / s

Dimension of ωt will be equal to the dimension of c.

$\sf:\implies\:[\omega][t]=[c]$

$\sf:\implies\:[T^{-1}][T^1]=[c]$

$\bf:\implies\:c=[T^0]=1$

∴ c is an unitless quantity.