The displacement from the position of equilibrium of a point 4 cm from a source of sinusoidal oscillations is half the amplitude at the moment `t=T//6` (T is the time perios). Assume that the source was at mean position at `t=0`. The wavelength of the running wave is :

Options

A. 0.96 m
B. 0.48 m
C. 0.24 m
D. 0.12 m

Answer

To solve the problem step by step, we will follow the reasoning and calculations presented in the video transcript.

### Step 1: Understand the Problem
We are given that the displacement from the equilibrium position of a point 4 cm from a source of sinusoidal oscillations is half the amplitude at the moment \( t = \frac{T}{6} \), where \( T \) is the time period. We need to find the wavelength of the running wave.

### Step 2: Write the General Equation of the Wave
The general equation for a sinusoidal wave can be expressed as:
\[ y(x, t) = A \sin(\omega t - kx) \]
where:
- \( y(x, t) \) is the displacement,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( x \) is the position,
- \( t \) is the time.

### Step 3: Substitute Known Values
We know that at \( t = \frac{T}{6} \) and \( x = 4 \, \text{cm} = 0.04 \, \text{m} \), the displacement \( y \) is half the amplitude:
\[ y = \frac{A}{2} \]
Substituting this into the wave equation gives:
\[ \frac{A}{2} = A \sin\left(\omega \frac{T}{6} - k(0.04)\right) \]
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \sin\left(\omega \frac{T}{6} - k(0.04)\right) \]

### Step 4: Determine Angular Frequency and Wave Number
The angular frequency \( \omega \) is given by:
\[ \omega = \frac{2\pi}{T} \]
Thus, substituting this into our equation:
\[ \frac{1}{2} = \sin\left(\frac{2\pi}{T} \cdot \frac{T}{6} - k(0.04)\right) \]
This simplifies to:
\[ \frac{1}{2} = \sin\left(\frac{\pi}{3} - k(0.04)\right) \]

### Step 5: Solve for the Angle
The sine of an angle equals \( \frac{1}{2} \) at:
\[ \theta = \frac{\pi}{6} \quad \text{or} \quad \theta = \frac{5\pi}{6} \]
Thus, we can set up the equation:
\[ \frac{\pi}{3} - k(0.04) = \frac{\pi}{6} \]
Solving for \( k(0.04) \):
\[ k(0.04) = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6} \]
Thus:
\[ k = \frac{\frac{\pi}{6}}{0.04} = \frac{\pi}{6 \times 0.04} = \frac{\pi}{0.24} \, \text{m}^{-1} \]

### Step 6: Relate Wave Number to Wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by:
\[ k = \frac{2\pi}{\lambda} \]
Setting the two expressions for \( k \) equal gives:
\[ \frac{\pi}{0.24} = \frac{2\pi}{\lambda} \]
Cancelling \( \pi \) from both sides:
\[ \frac{1}{0.24} = \frac{2}{\lambda} \]
Cross-multiplying gives:
\[ \lambda = 2 \times 0.24 = 0.48 \, \text{m} \]

### Final Answer
The wavelength of the running wave is:
\[ \lambda = 0.48 \, \text{m} \]