Question 99: Units , Dimensions and Error Analysis

Question

The speed of a wave produced in water is given by $v=\lambda^{a} g^{b} \rho^{c}$. Where $\lambda, g$ and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $a, b$ and $c$ respectively, are :

Accepted Answer

$$
\begin{aligned}
& v=\lambda^{\mathrm{a}} \mathrm{g}^{\mathrm{b}} ho^{\mathrm{c}} \\
& 	ext {using dimension formula} \\
& \Rightarrow\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-1}ight]=\left[\mathrm{L}^1ight]^{\mathrm{a}}\left[\mathrm{L}^1 \mathrm{~T}^{-2}ight]^{\mathrm{b}}\left[\mathrm{M}^1 \mathrm{~L}^{-3}ight]^{\mathrm{c}} \\
& \Rightarrow\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-1}ight]=\left[\mathrm{M}^{\mathrm{c}} \mathrm{L}^{\mathrm{a}+\mathrm{b}-\mathrm{c}} \mathrm{T}^{-2 \mathrm{~b}}ight] \\
& 	herefore \mathrm{c}=0, \mathrm{a}+\mathrm{b}-3 \mathrm{c}=1,-2 \mathrm{~b}=-1 \Rightarrow \mathrm{b}=\frac{1}{2} \\
& 	ext { Now } \mathrm{a}+\mathrm{b}-3 \mathrm{c}=1 \\
& \Rightarrow \mathrm{a}+\frac{1}{2}-0=1 \\
& \Rightarrow \mathrm{a}=\frac{1}{2} \\
& 	herefore \mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}, \mathrm{c}=0
\end{aligned}
$$

Answer

$\frac{1}{2}, 0, \frac{1}{2}$

Answer

$1,1,0$

Answer

$1,-1,0$

Answer

$\frac{1}{2}, \frac{1}{2}, 0$

Units , Dimensions and Error Analysis

The speed of a wave produced in water is given by v=\lambda^{a} g^{b} \rho^{c}. Where \lambda, g and \rho are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are :

Choose the correct answer: