Question 1:
moderate
In the given progressive wave equation y = 0.5 sin (10πt – 5x); where x, y in cm and t in second. The maximum velocity of the particle is
Question 2:
moderate
The path difference between the two waves
\[ y_{1}= a_{1} sin \left( \omega t -\frac{2\Pi x}{\lambda} \right) \]
and
\[ y_{2}= a_{2} cos \left( \omega t -\frac{2\Pi x}{\lambda} + \varphi \right) \]
is
Question 3:
The correct relation between frequencies of x-rays, \( \gamma\)-rays, heat rays and radio waves :
The electromagnetic spectrum in order of decreasing wavelength (and thus increasing frequency) is: Radio waves < Heat rays (Infrared) < X-rays \nu_x > \nu_{\text{heat}} > \nu_{\text{radio}}\).
Question 4:
easy
Two waves represented by the following equations are travelling in the same medium: \(y_1 = 5 \sin 2\pi(75t – 0.25x)\) and \(y_2 = 10 \sin 2\pi(150t – 0.50x)\). The intensity ratio \(I_1/I_2\) of the two waves is:
The intensity of a wave is proportional to the square of its amplitude and frequency, \(I \propto A^2 f^2\). Substituting \(A_1=5, f_1=75\) and \(A_2=10, f_2=150\) gives \(\frac{I_1}{I_2} = \left(\frac{5}{10}\right)^2 \left(\frac{75}{150}\right)^2 = \frac{1}{16}\).
Question 5:
easy
Equation of a progressive wave is given by \( y = 0.2 \cos \pi(0.04t + 0.02x – \pi/6) \). The distance is expressed in cm and time in second. What will be the minimum distance between two particles having the phase difference of \( \pi/2 \)?
The wave number is \(k = 0.02\pi\text{ cm}^{-1}\). Since phase difference \(\Delta \phi = k \Delta x\), we have \(\Delta x = \frac{\Delta \phi}{k} = \frac{\pi/2}{0.02\pi} = 25\text{ cm}\).
Question 6:
easy
A steel wire \(0.50\text{ m}\) long has a mass of \(4.0 \times 10^{-3}\text{ kg}\). If the wire is under a tension of \(80\text{ N}\), the speed of transverse waves on the wire is
Linear mass density \(\mu = \frac{m}{L} = \frac{4.0 \times 10^{-3}}{0.50} = 8.0 \times 10^{-3}\text{ kg/m}\). Wave speed is \(v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{80}{8.0 times 10^{-3}}} = 100\text{ m/s}\).
Question 7:
easy
A wave travelling in the positive x-direction having displacement amplitude along y-direction as 1 m, wavelength \(2\pi\text{ m}\) and frequency of \(\frac{1}{\pi}\text{ Hz}\) is represented by
Wave equation is \(y = A\sin(kx - \omega t)\). Here, \(A = 1\text{ m}\), \(k = \frac{2\pi}{\lambda} = 1\text{ m}^{-1}\), and \(\omega = 2\pi f = 2\text{ rad/s}\). Thus, \(y = \sin(x - 2t)\).
Question 8:
easy
Match List-I with List-II where list-I denotes nature and medium of wave and list-II denotes the expression for speed of wave. (All symbols have their usual meaning)
List-I
a. Transverse wave on a stretched string
b. Longitudinal wave in a metallic bar
c. Longitudinal wave in a fluid
List-II
(i) \( \sqrt{\frac{Y}{\rho}} \)
(ii) \( \sqrt{\frac{B}{\rho}} \)
(iii) \( \sqrt{\frac{T}{\mu}} \)
Choose the correct option:
Speed of a wave on a string is \( \sqrt{T/\mu} \) (a matches iii). Speed of a longitudinal wave in a metallic bar is \( \sqrt{Y/\rho} \) (b matches i). Speed of a longitudinal wave in a fluid is \( \sqrt{B/\rho} \) (c matches ii).
Question 9:
easy
Assertion (A): When a wave enters from one medium to another, its frequency is not changed.
Reason (R): Speed of a wave in a medium is property of the source.
The frequency of a wave is determined by the source that generates it and remains constant as the wave propagates from one medium to another. The speed of a wave, however, is a characteristic property of the medium it is traveling through, not the source. Thus, Assertion A is true, but Reason R is false.
Question 10:
easy
Assertion (A): Two waves moving in a uniform string having uniform tension cannot have different velocities.
Reason (R): Elastic and inertial properties of string are same for all waves in same string. Moreover, velocity of wave in a string depends on its elastic and inertial properties only.
The speed of a transverse wave on a string is given by \( v = \sqrt{T/\mu} \), where \( T \) is the tension (elastic property) and \( \mu \) is the linear mass density (inertial property). If the string is uniform (constant \( \mu \)) and has uniform tension (constant \( T \)), then \( v \) must be constant for all waves propagating on it. Both A and R are true, and R correctly explains A.