Question

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

• A. $y=a\sin \left(\frac{2\pi t}{T}\right)$
• B. $y=a\sin vt$
• C. $y=\frac{a}{T}\sin \left(\frac{t}{a}\right)$
• D. $y=a\sqrt{2}[\sin \left(\frac{2\pi t}{T}\right)-\cos \left(\frac{2\pi t}{T}\right)]$

Answer 1

Answer: (B), (C)

$y=\frac{a}{T}\sin \left(\frac{t}{a}\right)$

On the basis of “Principle of homogeneity” equal dimension can be added, subtracted or put equal to each other.

As y is the displacement of the particle, its unit should be equal to length, a is the amplitude of SHM and trigonometric functions are dimensionless.

(a) $y=a\sin \left(\frac{2\pi t}{T}\right)$

Here $\left(\frac{2\pi t}{T}\right)$ is dimensionless and $\left[y\right]=\left[a\right]$

(b) $y=a\sin vt$

Here $\left[vt\right]=\left[L^1\right]$
So trigonometric function is not dimensionless, hence this relation is not correct for simple harmonic motion.

(c) $y=\frac{a}{T}\sin \left(\frac{t}{a}\right)$
Here $\left(\frac{t}{a}\right)$ has dimensions so trigonometric function is not dimensionless, and $\left[y\right]\ne \left[\frac{a}{T}\right]$

Hence this relation is not correct for simple harmonic motion.

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

Here $\left(\frac{2\pi t}{T}\right)$ is dimensionless and $\left[y\right]=\left[a\right]$

Final Answer: (b),(c)