Question

Two particles of equal masses are revolving in circular paths of radii ${\mathrm{r}}_1$ and ${\mathrm{r}}_2$ respectively, with the same speed. The ratio of their centripetal forces is

• A. $\frac{r_2}{r_1}$
• B. ${\left(\frac{r_1}{r_2}\right)}^2$
• C. $\sqrt{\frac{r_2}{r_1}}$
• D. ${\left(\frac{r_2}{r_1}\right)}^2$

Answer 1

The correct option is A

$\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1}$

Step 1: Given data and assumptions made

The radius of the circular path for the first particle = ${\mathrm{r}}_1$

The radius of the circular path for the second particle = ${\mathrm{r}}_2$

Let, the mass of the two particles be $\mathrm{m}$ ; the speed of the particles be $\mathrm{v}$

Let, the centripetal force for one particle be ${\mathrm{F}}_1$ and the centripetal force for one particle be ${\mathrm{F}}_2$

Step 2: Explanation for the correct option

$\mathrm{centripetal}\mathrm{force},\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

As $\mathrm{m}$ and $\mathrm{v}$ are constants.

So, $\mathrm{F}\propto \frac{1}{\mathrm{r}}$

Therefore:

$$\begin{gathered} \frac{{\mathrm{F}}_1}{{\mathrm{F}}_2}=\frac{{\displaystyle \frac{1}{{\mathrm{r}}_1}}}{{\displaystyle \frac{1}{{\mathrm{r}}_2}}} \\ \frac{{\mathrm{F}}_1}{{\mathrm{F}}_2}=\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1} \end{gathered}$$

The ratio of their centripetal forces is $\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1}$, Thus, option A is correct.