$A$ book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:
$(a) \; y = a \sin \left(\frac{2 \pi t}{T}\right)$
$(b) \; y = a \sin v t$
$(c) \; y = \left(\frac{a}{T}\right) \sin \frac{t}{a}$
$(d) \; y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
($a =$ maximum displacement of the particle,$v =$ speed of the particle,$T =$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(B, C) Correct: $y = a \sin \left(\frac{2 \pi t}{T}\right)$
Dimension of $y = [L]$. Dimension of $a = [L]$. The argument of $\sin$ is $\frac{2 \pi t}{T}$,which is dimensionless. Thus,the formula is dimensionally correct.
$(b)$ Incorrect: $y = a \sin v t$
Dimension of $v t = [LT^{-1}] \times [T] = [L]$. The argument of the trigonometric function must be dimensionless,but here it has dimensions of length. Thus,it is dimensionally incorrect.
$(c)$ Incorrect: $y = \left(\frac{a}{T}\right) \sin \left(\frac{t}{a}\right)$
Dimension of $\frac{a}{T} = [LT^{-1}] \neq [L]$. Also,the argument $\frac{t}{a} = [TL^{-1}]$ is not dimensionless. Thus,it is dimensionally incorrect.
$(d)$ Correct: $y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
Dimension of $y = [L]$. Dimension of $a = [L]$. The argument $\frac{2 \pi t}{T}$ is dimensionless. Thus,the formula is dimensionally correct.

Explore More

Similar Questions

The equation for a real gas is given by $(P + \frac{a}{V^2})(V - b) = RT$,where $P, V, T$ and $R$ are the pressure,volume,temperature,and gas constant,respectively. The dimension of $ab^{-2}$ is equivalent to that of

What could be a correct expression for the speed of ocean waves in terms of its wavelength $\lambda$,the depth $h$ of the ocean,the density $\rho$ of sea water,and the acceleration of free fall $g$?

The equation of motion of a damped oscillator is given by $m \frac{d^2 x}{d t^2}+b \frac{d x}{d t}+k x=0$. The dimensional formula of $\frac{b}{\sqrt{k m}}$ is

In the expression $A=B+\frac{C}{D+E}$,the dimensions of physical quantities $B$ and $C$ are $[L^{1} M^{0} T^{-1}]$ and $[L^{1} M^{0} T^{0}]$ respectively. The dimensions of quantities $A, D$ and $E$ are

If energy $(E)$,velocity $(V)$,and time $(T)$ are chosen as the fundamental quantities,the dimensional formula of surface tension will be

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo