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 dimensional analysis an equation must be dimensionally homoheneous.
(a) `y = a "sin" (2pi t)/T`
Here, `[L.H.S.]=[y]=[L]`
and `[R.H.S.]=[a "sin" (2 pi t)/T]=[L "sin"T/T]=[L]`
So, it is correct.
(b) `y = a sin vt`
Here, `[y]=[L]` and `[a sin vt]=[L sin (LT^(-1).T)]=[L sin L]`
so, the equation is wrong.
(c) `y = (a/T)"sin" t/a`
Here, `[y]=[L]` and `[(a/T)"sin" t/a]=[L/T"sin"T/L]=[LT^(-1) sin TL^(-1)]`
So, the equation is wrong.
(d) `y=(asqrt(2))("sin"(2pit)/T+"cos"(2pit)/T)`
Here, `[y]=[L], [asqrt(2)]=[L]`
and `["sin"(2pit)/T+"cos"(2pit)/T]=["sin"T/T+ "cos"T/T]=` dimensionless
So, the equation is correct.