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]

Question

Smart Achievers · Accepted Answer

(1)

Particle velocity $10 cm/s$ 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.

Solution

1 $\frac{\sqrt{3} π}{50} \hat{j} m/s$

2 $- \frac{\sqrt{3} π}{50} \hat{j} m/s$

3 $\frac{\sqrt{3} π}{50} \hat{i} m/s$

4 $- \frac{\sqrt{3} π}{50} \hat{i} m/s$

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Solution

1 3π50 j^ m/s

2 -3π50 j^ m/s

3 3π50 i^ m/s

4 -3π50 i^ m/s

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1 $\frac{\sqrt{3} π}{50} \hat{j} m/s$

2 $- \frac{\sqrt{3} π}{50} \hat{j} m/s$

3 $\frac{\sqrt{3} π}{50} \hat{i} m/s$

4 $- \frac{\sqrt{3} π}{50} \hat{i} m/s$