The speed of a water wave v depends on its wa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the relationship be $v = k g^a \lambda^b \rho^c$,where $k$ is a dimensionless constant.Writing the dimensions of each quantity:$[v] = [L T^{-1

Vedclass · Accepted Answer

$v^2 \propto g \lambda$

Vedclass · Answer

(C) Let the relationship be $v = k g^a \lambda^b \rho^c$,where $k$ is a dimensionless constant.Writing the dimensions of each quantity:$[v] = [L T^{-1}]$$[g] = [L T^{-2}]$$[\lambda] = [L]$$[\rho] = [M L^{-3}]$Substituting these into the equation: $[L T^{-1}] = [L T^{-2}]^a [L]^b [M L^{-3}]^c$$[L^1 T^{-1} M^0] = [M^c L^{a+b-3c} T^{-2a}]$Comparing the powers of $M, L,$ and $T$ on both sides:For $M$: $c = 0$For $T$: $-2a = -1 \implies a = 1/2$For $L$: $a + b - 3c = 1 \implies 1/2 + b - 0 = 1 \implies b = 1/2$Thus,$v \propto g^{1/2} \lambda^{1/2} \rho^0 \implies v \propto \sqrt{g \lambda} \implies v^2 \propto g \lambda$.

The speed of a water wave $v$ depends on its wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. Determine the relationship between these quantities.

Similar Questions

From the following combinations of physical constants (expressed through their usual symbols),the only combination that would have the same value in different systems of units is:

If the force is given by $F = at + bt^2$ where $t$ is time, the dimensions of $a$ and $b$ are:

Convert $1 \; \text{newton}$ ($SI$ unit of force) into $dyne$ ($CGS$ unit of force).

The density of a material in the $CGS$ system of units is $4\,g/cm^3$. In a system of units in which the unit of length is $10\,cm$ and the unit of mass is $100\,g$,the value of the density of the material will be:

In a new system of units,energy $(E)$,density $(d)$,and power $(P)$ are taken as fundamental units. Then the dimensional formula of the universal gravitational constant $G$ will be .......

Vedclass Test Series

Exam Paper Generator

Online Exam Module