Given:
- Initial resonating length, \( L = 90 \, \text{cm} \)
- Initial fundamental frequency, \( f_0 = 400 \, \text{Hz} \)
- New fundamental frequency, \( f' = 600 \, \text{Hz} \)
Step 1: Relation Between Frequency and Length
The fundamental frequency of a vibrating string is given by:
\[ f_0 = \frac{v}{2L}, \]
where:
- \( v \) is the wave speed,
- \( L \) is the length of the wire.
For the same tension, the wave speed \( v \) remains constant.
Step 2: Expressing New Length in Terms of Frequency
Let the new resonating length be \( L' \) for the frequency \( f' \). The new fundamental frequency is given by:
\[ f' = \frac{v}{2L'}. \]
Dividing the two equations:
\[ \frac{f'}{f_0} = \frac{L}{L'}. \]
Rearranging to find \( L' \):
\[ L' = L \times \frac{f_0}{f'}. \]
Step 3: Substituting the Given Values
Substituting the values:
\[ L' = 90 \times \frac{400}{600}. \]
Simplifying:
\[ L' = 90 \times \frac{2}{3} = 60 \, \text{cm}. \]
Therefore, the new resonating length of the wire is \( 60 \, \text{cm} \).
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\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
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