Problem 7 on Units & Dimensions

Description:

Problem

A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion.

• y = a sin 2πt/T

• y = a sin vt

• y = (a/T) sin t/a

• y = (a√2) (sin 2πt/T + cos 2πt/T)

(a represents maximum displacement,

v represents speed of the particle,

T represents time period of motion)

Rule out wrong formulas on dimensional grounds.

Solution

Before we start solving the problem we should recap the following rules of dimensional consistency −

• A ± B, valid only when [A] = [B]

• A = B, valid equation only when [A] = [B]

• Quantities inside the arguments of trigonometric, exponential or logarithmic functions have to be dimensionless.

Part 1

y = a sin 2πt/T

Dimension of 2πt/T = [T]/[T] = 1, Argument of sine function is dimensionless.

Dimension of L.H.S ([L]) = Dimension of R.H.S. ([L])

Hence, y = a sin 2πt/T, is Dimensionally consistent.

Part 2

y = a sin vt

Dimension of vt = [L]/[T] × [T] = [L],

Argument of sine function is NOT dimensionless.

Hence, y = a sin vt, is Dimensionally NOT consistent.

Part 3

y = (a/T) sin t/a

Dimension of vt = [T]/[L] = [T][L]^-1,

Argument of sine function is NOT dimensionless.

Hence, y = (a/T) sin t/a, is Dimensionally NOT consistent.

Part 4

y = (a√2) (sin 2πt/T + cos 2πt/T)

Dimension of 2πt/T = [T]/[T] = 1, Argument of sine and cosine are dimensionless,

Dimension of L.H.S ([L]) = Dimension of R.H.S.([L])

Hence, y = (a√2) (sin 2πt/T + cos 2πt/T), is Dimensionally consistent.