Experiments reveal that the velocity `v` of water waves may depend on their wavelength `lambda`, density of water `rho`, and acceleration due to gravity `g`. Establish a possible relation between `v and lambda , g, rho`.
Text Solution
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To establish a possible relation between the velocity \( v \) of water waves and the wavelength \( \lambda \), density of water \( \rho \), and acceleration due to gravity \( g \), we can use dimensional analysis. Here’s a step-by-step solution:
### Step 1: Identify the dimensions of each variable
- **Velocity \( v \)**: The dimension of velocity is \( [v] = L T^{-1} \).
- **Wavelength \( \lambda \)**: The dimension of wavelength is \( [\lambda] = L \).
- **Density \( \rho \)**: The dimension of density is \( [\rho] = M L^{-3} \).
- **Acceleration due to gravity \( g \)**: The dimension of acceleration is \( [g] = L T^{-2} \).
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