A book with many printing errors contains four different forumlae for the displacement y of a particle undergoing a certain periodic motion : (i) `y = a sin (2pi t)/(T)` (ii) `y = a sin upsilon t` (iii) `y = (a)/(T) sin (t)/(a)` (iv) `y= (a)/(sqrt2)[sin(2pi t)/(T) + cos (2pi t)/(T)]` Here, a is maximum displacement of particle, `upsilon` is speed of particle, T is time period of motion. Rule out the wrong forumlae on dimensinal grounds.

Explanation

The argument of a trigonmietrical funciton, i.e., angle is dimensionless. Now, in
(i) `(2pi t)/(T) = (T)/(T) = 1 =(M^0 L^0 T^0)` …. "Dimensionless"`
(ii) `upsilon t = (LT^(-1))(T) = L = [M^0 L^1 T^0] …." Not dimensionless"`
(iii) `(t)/(a) = (T)/(L) = [L^(-1) T^1] ....." not dimensionless"`
(iv) `(2pit)/(T) = (T)/(T) = 1 = [M^0L^0 T^0] ...."dimensionless"`
:. formulea (ii) and (iii) are wrong.