The equation of a standing wave, produced on a string fixed at both ends, is ` y = (0.4 cm) sin[(0.314 cm^-1) x] cos[(600pis^-1)t]` What could be the smallest length of the string ?

Explanation

To find the smallest length of the string that produces the standing wave described by the equation \( y = (0.4 \, \text{cm}) \sin[(0.314 \, \text{cm}^{-1}) x] \cos[(600 \pi \, \text{s}^{-1}) t] \), we can follow these steps:

### Step 1: Identify the wave number \( k \)
From the given equation, we can see that the wave number \( k \) is given as \( 0.314 \, \text{cm}^{-1} \).

### Step 2: Calculate the wavelength \( \lambda \)
The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{0.314} \]
Calculating this gives:
\[ \lambda \approx \frac{6.28}{0.314} \approx 20 \, \text{cm} \]

### Step 3: Determine the smallest length of the string
For a string fixed at both ends, the smallest length \( L \) that can support a standing wave is given by:
\[ L = \frac{\lambda}{2} \]
Substituting the calculated value of \( \lambda \):
\[ L = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Conclusion
Thus, the smallest length of the string is:
\[ \boxed{10 \, \text{cm}} \]

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