On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:

a) y=a\; \sin \; \frac{2\pi t}{T}

b) y=a\; \sin \; vt

c) y=\frac{a}{T}\sin (\frac{t}{a})

d) y=a\sqrt{2}[\sin (\frac{2\pi t}{t})-\cos (\frac{2\pi t}{T})]

Answers (1)

The answer is the option  (b) y=a\; \sin \; vt and  (c) y=\frac{a}{T}\sin (\frac{t}{a})

Explanation: (b) Here, v.t is an angle, whose dimensions are – [LT^{-1}] [T] = [L]

Thus, it is not true that sin vt is dimensionless.

(c) Here the dimension of amplitude a/T on the R.H.S. is equal to \frac{[L]}{[T]} = [LT^{-1}], viz., not equal to the dimensions of y and the angle \frac{t}{a} = [LT^{-1}], which means that they are not dimensionless.

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