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A book with many printing errors contain...

A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :- `y=a sin 2pi t//T` (a = maximum displacement of the particle, v = speed of the particle. T= time-period of motion). Rule out the wrong formulas on dimensional grounds.

Text Solution

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The argument of trigonometric function i.e. angle is dimensionless.
From principle of homogeneity,
(a) `y=asin2pit//Timplies (2pit)/(T)=([T])/([T])=1`, dimensionless. `[y]=[a]`.
So it is correct.
(b) `y=asinvtimpliesvt=[LT^(-1)][T]=[L]`, `:.vt` is not dimensionless. So it is wrong.
(c ) `y=(a//T)sint//aimplies(t)/(a)=([T])/([LT^(-2)])=[L^(-1)T^(3)]` `:.(t)/(a)` is not dimensionless. So it is wrong.
(d) `y=(asqrt(2))(sin2pit//T+cos2pit//T)implies(2pit)/(T)=([T])/([T])=1`
`(2pit)/(T)` is dimensonless. So it is correct
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