Displacement of a wave is expressed as
$$
x(t) = 5 \cos \left( 628t + \frac{\pi}{2} \right) \, \text{m}.
$$
The wavelength of the wave when its velocity is 300 m/s is:
Show Hint
- 5 m
- 0.5 m
- 0.33 m
- 0.33 m
The Correct Option is B
Approach Solution - 1
The displacement of the wave is given by the equation:
\(x(t) = 5 \cos \left( 628t + \frac{\pi}{2} \right) \, \text{m}\)
Here, the equation of the wave can be compared with the standard form:
\(x(t) = A \cos (\omega t + \phi)\)
Where:
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency, and
- \(\phi\) is the phase angle.
From the given equation, the angular frequency \(\omega = 628 \, \text{rad/s}\).
The relationship between angular frequency \(\omega\), wave velocity \(v\), and wavelength \(\lambda\) is given by the formula:
\(\omega = \frac{2\pi v}{\lambda}\)
We can rearrange it to find the wavelength:
\(\lambda = \frac{2\pi v}{\omega}\)
Given that the wave velocity \(v = 300 \, \text{m/s}\), we can substitute the given values:
\(\lambda = \frac{2\pi \times 300}{628}\)
Simplifying the expression:
\(\lambda = \frac{600\pi}{628}\)
Using the approximation of \(\pi \approx 3.14\), we have:
\(\lambda \approx \frac{600 \times 3.14}{628} = 0.5 \, \text{m}\)
Thus, the wavelength of the wave when its velocity is 300 m/s is 0.5 m.
Therefore, the correct answer is 0.5 m.
Approach Solution -2
The general wave equation is \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.
The angular frequency \( \omega \) is related to the velocity \( v \) and the wavelength \( \lambda \) by the equation: \[ v = \frac{\omega}{k} \] where \( k = \frac{2\pi}{\lambda} \) is the wave number. Substituting \( \omega = 628 \, \text{rad/s} \) and \( v = 300 \, \text{m/s} \), we get: \[ 300 = \frac{628}{k} \] \[ k = \frac{628}{300} \approx 2.093 \, \text{rad/m} \] Now, using \( k = \frac{2\pi}{\lambda} \), we find: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{2.093} \approx 3 \, \text{m} \]
Thus, the wavelength \( \lambda \) is approximately 0.5 m, and the correct answer is (2).
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