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Question : 14 of 33
Marks: +1, -0
book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion
(a) y=asin⁡2πtTy = a \sin \frac{2\pi t}{T}
(b) y=asin⁡vty = a \sin vt
(c) y=aTsin⁡tay = \frac{a}{T} \sin \frac{t}{a}
(d)y=(a2)(sin⁡2πtT+cos⁡2πtT)y = (a \sqrt{2}) \left( \sin \frac{2\pi t}{T} + \cos \frac{2\pi t}{T} \right)
(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion.)
Rule out the wrong formulas on dimensional grounds.
Solution:  
The argument of a trigonometrical function, i.e. angle isdimensionless. Now using the principle of homogeneity of dimensions.
(a) 2πtT=[T][T]=1,\frac{2\pi t}{T} = \frac{[T]}{[T]} = 1, dimensionless.
(b) vt=[LT−1][T]=[L],v t = [LT^{-1}] [T] = [L], dimension of length.
(c) ta=[T][LT−2]=[L−1T3],\frac{t}{a} = \frac{[T]}{[LT^{-2}]} = [L^{-1} T^{3}], not dimensionless.
(d) 2πtT=[T][T]=1,\frac{2\pi t}{T} = \frac{[T]}{[T]} = 1, dimensionless.
Hence (b) and (c) are wrong on dimensional grounds.
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