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

According to the principle of homogeneity,
The dimensions on left side of an equation = The dimensions on the right side of an equation. Also the arguments of sine or cosine function must be dimensionless dimensionally.
(a) The relation is correct because `2pi t"/"T=T"/"T=1` (dimensionless) and the dimensions of y and a are same.
(b) Dimensions of `ut=[L^(1)T^(-1)xxT^(1)]=[L^1]`. But ut should be dimensionless. Therefore, the relation is wrong.
(c ) The relation is not correct because y and `(a"/"d)` have different dimensions and `t"/"a` is not dimensionless.
(d) Here the angle `(2 pi t)/(T)` is dimensions and the dimensions on both sides of the equation are equal. Therefore, it is dimensionally correct.