Question 426: Waves

Question

A transverse wave given by $$y=2 \sin (0.01 x+30 t)$$ moves on a stretched string from one end to another end in 0.5 second. If '$$x$$' and '$$y$$' are in $$\mathrm{cm}$$ and '$$\mathrm{t}$$' is in second, then the length of the string is

Accepted Answer

To solve this problem, we need to find the length of the string over which the transverse wave described by the equation $$y = 2 \sin(0.01x + 30t)$$ travels in 0.5 seconds.
The general form of a transverse wave equation is:
$$y(x, t) = A \sin(kx - \omega t + \phi)$$
Where:
$$A$$ is the amplitude of the wave.
$$k$$ is the wave number, given by $$k = \frac{2 \pi}{\lambda}$$, where $$\lambda$$ is the wavelength.
$$\omega$$ is the angular frequency, given by $$\omega = 2 \pi f$$, where $$f$$ is the frequency.
$$\phi$$ is the phase constant.
From the given wave equation $$y = 2 \sin(0.01x + 30t)$$:
The coefficient of $$x$$ is the wave number $$k$$, so $$k = 0.01$$ cm
-1
.
The coefficient of $$t$$ is the angular frequency $$\omega$$, so $$\omega = 30$$ rad/s.
The wave velocity $$v$$ can be calculated using the relationship between the angular frequency and the wave number:
$$v = \frac{\omega}{k}$$
Substituting the given values:
$$v = \frac{30}{0.01} = 3000$$ cm/s
The time taken for the wave to travel the length of the string is given as 0.5 seconds. Therefore, the length of the string $$L$$ can be found using the formula:
$$L = v \cdot t$$
Substituting the given values:
$$L = 3000 	imes 0.5 = 1500$$ cm
Converting this into meters:
$$L = 1500 \, 	ext{cm} 	imes \frac{1 \, 	ext{m}}{100 \, 	ext{cm}} = 15 \, 	ext{m}$$
Therefore, the length of the string is 15 meters.
The correct option is:
Option D: 15 m

Answer

6 m

Answer

9 m

Answer

12 m

Answer

15 m

Waves

A transverse wave given by $y=2 \sin (0.01 x+30 t) moves on a stretched string from one end to another end in 0.5 second. If 'x' and 'y' are in \mathrm{cm} and '\mathrm{t}$' is in second, then the length of the string is

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