Question

Check the correctness of the equation, $\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$ where,$\mathrm{F}$ is force, $\mathrm{m}$ is mass, $\mathrm{v}$ is velocity, and $\mathrm{r}$ is radius.

Answer 1

Step 1: Concept used

Principle of homogeneity:

1. The principle of homogeneity of dimensions states that an equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same.
2. The principle of Homogeneity states that the dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another.

Step 2: Given parameters

The equation given is,

$\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}...\left(1\right)$

Where, $\mathrm{F}$ is the force, $\mathrm{m}$ is the mass, $\mathrm{v}$ is the velocity, and, $\mathrm{r}$ is the radius.

Step 3: Correctness of the equation

From Newton's first law of motion, we know that,

$\mathrm{F}=\mathrm{ma}...\left(2\right)$

Comparing equation$\left(1\right)$ and equation $\left(2\right)$ as follows:

$\mathrm{ma}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

Write in terms of dimensional formula.

${\mathrm{MLT}}^{-2}={\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{L}}^{-1}$

${\mathrm{MLT}}^{-2}={\mathrm{MLT}}^{-2}$

By the principle of homogeneity, the equation is correct.

Hence, $\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$is the correct equation.