Which of the following relations for the displacement of a partide undergoing simple harmonic motion is not correct dimensionally?

Y = a sin \frac{2 π t}{T}

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Y = a cos ω t

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Y = \frac{a}{T} sin (\frac{t}{a})

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Y = a \sqrt{2} (sin \frac{2 π t}{T} + cos \frac{2 π t}{T})

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Solution

The correct option is D $Y = \frac{a}{T} sin (\frac{t}{a})$
$\begin{matrix} Y = \frac{a}{T} sin (\frac{t}{a}) N o w, a \to m e t e r s T \to s e c o n d s a / T = m / s \to u n i t o f v e l o c i t y \end{matrix}$

Dimensions on RHS must be displacement [L]. Arguments of sine and cosine are dimensionless. Hence, the option (c) is not correct.

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Similar questions

A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:

(a)

(b) y = a sin vt

(c)

(d)

(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

y = A sin (V t)

y = \frac{A}{T} sin (\frac{t}{A})

y = \frac{A}{\sqrt{2}} (s i n (w t) + c o s (w t))

Y

Y = 2 sin t {cos}^{2} \frac{t}{2}

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