The speed of a wave produced in water is given by . When , g and are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

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Solution

Q.
The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

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

2 1, 1, 0

3 1, –1, 0

4 $\frac{1}{2}, 0, \frac{1}{2}$

Ans.
(1)

$v = λ^{a} g^{b} ρ^{c}$

using dimension formula

$\Rightarrow [M^{0} L^{1} T^{- 1}] = {[L^{1}]}^{a} {[L^{1} T^{- 2}]}^{b} {[M^{1} L^{- 3}]}^{c}$

$\Rightarrow [M^{0} L^{1} T^{- 1}] = [M^{c} L^{a + b - 3 c} T^{- 2 b}]$

$∴ c = 0, a + b - 3 c = 1, - 2 b = - 1 \Rightarrow b = \frac{1}{2}$

Now, a + b –3c = 1

$\Rightarrow a + \frac{1}{2} - 0 = 1$

$\Rightarrow a = \frac{1}{2}$

$∴ a = \frac{1}{2}, b = \frac{1}{2}, c = 0$

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