6. (JEE Main 2022 (Online) 26th July Evening Shift)

A transverse wave is represented by $y = 2 \sin (ω t - k x) cm$ . The value of wavelength (in $cm$ ) for which the wave velocity becomes equal to the maximum particle velocity, will be :

A. 4 $π$

B. 2 $π$

C. $π$

D. 2

Correct Answer is Option (A)

$\frac{ω}{k} = A ω$

$\Rightarrow k = \frac{1}{A} = \frac{1}{2 c m}$

$\Rightarrow \frac{2 π}{λ} = \frac{1}{2 c m}$

$\Rightarrow λ = 4 π c m$

7. (JEE Main 2022 (Online) 30th June Morning Shift)

Which of the following equations correctly represents a travelling wave having wavelength $λ$ = 4.0 cm, frequency v = 100 Hz and travelling in positive x-axis direction?

A. $y = A \sin [(0.50 π c m^{- 1}) x - (100 π s^{- 1}) t]$

B. $y = A \sin 2 π [(0.25 c m^{- 1}) x - (50 s^{- 1}) t]$

C. $y = A \sin [(\frac{2 π}{4} c m^{- 1}) x - (\frac{2 π}{100} s^{- 1}) t]$

D. $y = A \sin π [(0.5 c m^{- 1}) x - (200 s^{- 1}) t]$

Correct Answer is Option (D)

We know, equation of wave travelling in positive x-direction is -

$y = A \sin (k x - w t)$

where $k = \frac{2 π}{λ}$

and $w = 2 π f$

Here given $λ$ = 4 cm and frequency (f) = 100 Hz

$∴$ $k = \frac{2 π}{4} = 0.5 π$ cm^{$-$ 1}

and $w = 2 π \times 100 = 200 π$ s^{$-$ 1}

$∴$ Equation of travelling wave,

$y = A \sin (0.5 π x - 200 π t)$

8. (JEE Main 2022 (Online) 29th June Morning Shift)

A longitudinal wave is represented by $x = 10 \sin 2 π (n t - \frac{x}{λ})$ cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to :

A. 2 $π$

B. 5 $π$

C. $π$

D. $\frac{5 π}{2}$

Correct Answer is Option (B)

Particle velocity = $\frac{\partial x}{\partial t}$

$\Rightarrow$ Maximum particle velocity $= (2 π n) (10)$

$\Rightarrow (2 π n) (10) = (n λ) (4)$

$\Rightarrow λ = 5 π$

9. (JEE Main 2022 (Online) 27th June Evening Shift)

If a wave gets refracted into a denser medium, then which of the following is true?

A. wavelength, speed and frequency decreases.

B. wavelength increases, sped decreases and frequency remains constant.

C. wavelength and speed decreases but frequency remains constant.

D. wavelength, speed and frequency increases.

Correct Answer is Option (C)

Frequency is independent of medium. For denser medium, wavelength and speed both would decrease.

10. (JEE Main 2021 (Online) 22th July Evening Shift)

The motion of a mass on a spring, with spring constant K is as shown in figure.

JEE Main 2021 (Online) 22th July Evening Shift Physics - Waves Question 44 English
The equation of motion is given by
x(t) = A sin $ω$ t + B cos $ω$ t with $ω$ = $\sqrt{\frac{K}{m}}$

Suppose that at time t = 0, the position of mass is x(0) and velocity v(0), then its displacement can also be represented as x(t) = C cos( $ω$ t $-$ $ϕ$ ), where C and $ϕ$ are :

A. $C = \sqrt{\frac{2 v {(0)}^{2}}{ω^{2}} + x {(0)}^{2}}, ϕ = \tan^{- 1} (\frac{x (0) ω}{2 v (0)})$

B. $C = \sqrt{\frac{v {(0)}^{2}}{ω^{2}} + x {(0)}^{2}}, ϕ = \tan^{- 1} (\frac{x (0) ω}{v (0)})$

C. $C = \sqrt{\frac{v {(0)}^{2}}{ω^{2}} + x {(0)}^{2}}, ϕ = \tan^{- 1} (\frac{v (0)}{x (0) ω})$

D. $C = \sqrt{\frac{2 v {(0)}^{2}}{ω^{2}} + x {(0)}^{2}}, ϕ = \tan^{- 1} (\frac{v (0)}{x (0) ω})$

Correct Answer is Option (C)

C \cos ϕ = x (0)

t C ω \sin ϕ = v (0)

{[\frac{v (0)}{ω}]}^{2} + [x (0)]^{2} = C^{2}

\tan ϕ = \frac{v (0)}{x (0) ω}