The equation of a standing wave, set up in a string is, y=0.8 sin[(0.314`cm^(-1)`)x]cos[(1200`pis^(-1)`)t]. Calculate the smallest possible length of the living.

Answer

To solve the problem, we need to analyze the given equation of the standing wave and find the smallest possible length of the string. The equation provided is:

\[ y = 0.8 \sin(0.314 \, \text{cm}^{-1} \, x) \cos(1200 \, \pi \, \text{s}^{-1} \, t) \]

### Step 1: Identify the wave number \( k \)
The wave number \( k \) can be identified from the equation of the standing wave. In the equation, the term associated with \( x \) is \( 0.314 \, \text{cm}^{-1} \), which means:

\[ k = 0.314 \, \text{cm}^{-1} \]

### Step 2: Calculate the wavelength \( \lambda \)
The relationship between the wave number \( k \) and the wavelength \( \lambda \) is given by:

\[ k = \frac{2\pi}{\lambda} \]

Rearranging this gives:

\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):

\[ \lambda = \frac{2\pi}{0.314} \]

Calculating \( \lambda \):

\[ \lambda \approx \frac{6.2832}{0.314} \approx 20 \, \text{cm} \]

### Step 3: Determine the length of the string
The smallest possible length of the string that can support a standing wave is equal to half the wavelength (since the fundamental mode of vibration has one antinode at each end). Therefore, the length \( L \) of the string is given by:

\[ L = \frac{\lambda}{2} \]

Substituting the value of \( \lambda \):

\[ L = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Final Answer
The smallest possible length of the string is:

\[ L = 10 \, \text{cm} \]

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