JEE Advanced PYQ Basic of Mechanical Waves, Progressive and Stationary Waves

Q 1 :

Two monoatomic ideal gases 1 and 2 of molecular masses $m_{1}$ and $m_{2}$ respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2 is given by [2000]

$\sqrt{\frac{m_{1}}{m_{2}}}$
$\sqrt{\frac{m_{2}}{m_{1}}}$
$\frac{m_{1}}{m_{2}}$
$\frac{m_{2}}{m_{1}}$

1

2

3

4

(2)

$V = \sqrt{\frac{γ R T}{M_{0}}},$ $where M_{0} = molecular mass$

$So, V \propto \sqrt{\frac{1}{M_{0}}} \Rightarrow \frac{V_{1}}{V_{2}} = \sqrt{\frac{M_{02}}{M_{01}}} = \sqrt{\frac{m_{2}}{m_{1}}}$

Q 2 :

A transverse sinusoidal wave moves along a string in the positive $x$ -direction at a speed of $10 cm/s$ . The wavelength of the wave is $0.5 m$ and its amplitude is $10 cm$ . At a particular time $t$ , the snapshot of the wave is shown in figure. The velocity of point $P$ when its displacement is $5 cm$ is [2008]

$\frac{\sqrt{3} π}{50} \hat{j} m/s$
$- \frac{\sqrt{3} π}{50} \hat{j} m/s$
$\frac{\sqrt{3} π}{50} \hat{i} m/s$
$- \frac{\sqrt{3} π}{50} \hat{i} m/s$

1

2

3

4

(1)

Particle velocity $v_{p}$ is related to the displacement of the particle from the mean position as

$v_{p} = 2 π ν \sqrt{A^{2} - y^{2}}$

$v_{p} = 2 π (\frac{v}{λ}) \sqrt{A^{2} - y^{2}}$

$= \frac{2 π}{0.5} \times 0.1 \sqrt{{(0.1)}^{2} - {(0.05)}^{2}} = \frac{\sqrt{3 π}}{50} \hat{j} m/s$

Since the wave is sinusoidal moving in positive $x$ -axis, the point will move parallel to $y$ -axis therefore options (3) and (4) are ruled out. As the wave moves forward in positive $X$ -direction, the point should move upwards i.e. in the positive $Y$ -direction.

Q 3 :

A block M hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength $λ_{0}$ is produced at point O on the rope. The pulse takes time $T_{O A}$ to reach point A. If the wave pulse of wavelength $λ_{0}$ is produced at point A (Pulse 2) without disturbing the position of M, it takes time $T_{A O}$ to reach point O. Which of the following options is/are correct? [2017]

The time $T_{A O} = T_{O A}$
The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope
The wavelength of Pulse 1 becomes longer when it reaches point A
The velocity of any pulse along the rope is independent of its frequency and wavelength

1

2

3

4

Select one or more options

(1, 4)

$Wavelength of pulse, λ = \frac{v}{f} = \frac{1}{f} \sqrt{\frac{T}{μ}} or, T \propto \sqrt{T}$

Where $T =$ tension of string.

Here $T_{1} > T_{2} ∴ λ_{1} > λ_{2}$

The velocities of the two pulses cannot be same at mid-point as velocity being vector quantity has direction.

$V = \sqrt{\frac{T}{μ}},$ so speed at any position will be same for both pulses, therefore time taken by both pulses will be same i.e., $T_{A O} = T_{O A}$

Topic Question Set

Q 1 :

Two monoatomic ideal gases 1 and 2 of molecular masses $m_{1}$ and $m_{2}$ respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2 is given by [2000]

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Topic Question Set

Q 1 : Two monoatomic ideal gases 1 and 2 of molecular masses m1 and m2 respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2 is given by [2000]

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Q 1 :

Two monoatomic ideal gases 1 and 2 of molecular masses $m_{1}$ and $m_{2}$ respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2 is given by [2000]