Question

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

(a) y = a sin 2πt/T

(b) y = a sin vt

(c) y = (a/T) sin t/a

(d) y = (a√2) (sin 2πt/T + cos 2πt/T)

(a = maximum displacement of the particle, b a speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds. 

Answer 1

Here we need the dimension of the LHS = dimension of RHL = L¹

Also, the dimension of the argument of the trigonometric function

= dimensionless

(a) Dimension of the argument of the sine function

(T¹/T¹) = T⁰ = dimensionless.

Dimension of a = L¹ = dimensions of y.

∴ This formula is dimensionally correct

(b) Dimension of the argument of the sine function

= [LT^-1] [T¹] = L¹

∴ This formula is dimensionally incorrect

(c) Dimension of the argument of the sine function

= t/a = T/L = [L^-1] [T¹] Also,

dimension of (a/t) = L/T = LT^-1 Clearly,

∴ This formula is dimensionally incorrect.

(d) Dimension of a√2 = L¹ = dimension of y Dimension of the argument of the sine function

= dimension of the argument of the cosine function = 2π(t/T) = T¹/T¹ 

= T° = dimensionless.

∴ This formula is dimensionally correct.