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

A. y = a sin 2 t/T

B. y = a sin vt

C. y = (a/T) sin t/a

D.

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

The dimensions of the quantities involved are:

y = M⁰L¹T⁰

a = M⁰L¹T⁰

v = M⁰L¹T^-1

T or t = M⁰L⁰T¹

A. correct

Explanation: t/T is unitless. Therefore, dimension of y should be the dimension of a which is true.

B. incorrect

Explanation: quantities inside the trigonometric expressions should be unitless but dimension of vt = M⁰L¹T⁰ which is having unit of length and is therefore wrong.

C. incorrect

Explanation: quantities inside the trigonometric expressions should be unitless but dimension of t/a = M⁰L^-1T¹ which is having unit of time/length and is therefore wrong.

D. correct

Explanation: t/T is unitless. Therefore, dimension of y should be the dimension of a which is true.