Question

Two wheels of radii r₁ and r₂ and circumferences c₁ and c₂ respectively. Cover the same distance in same time. If in that time, they complete n₁ and n₂ revolutions respectively.
Prove that $\frac{n_2}{n_1}=\frac{r_1}{r_2}=\frac{c_1}{c_2}$

Answer 1

Radius of the 1st wheel = r₁
Radius of the 2nd wheel = r₂
Circumference of the 1st wheel = c₁
Circumference of the 2nd wheel = c₂
Number of revolution taken by the 1st wheel = n₁
Number of revolution taken by the 2nd wheel = n₂
Total distance covered by the 1st wheel = $n_1\times c_1$
Total distance covered by the 2nd wheel = $n_2\times c_2$
Total distance covered by the 1st wheel = total distance covered by the 2nd wheel
$$\begin{gathered} \Rightarrow n_1\times c_1=n_2\times c_2 \\ \Rightarrow \frac{c_1}{c_2}=\frac{n_1}{n_2}......\left(1\right) \\ \Rightarrow \frac{c_1}{c_2}=\frac{2\pi r_1}{2\pi r_2} \\ \Rightarrow \frac{c_1}{c_2}=\frac{r_{\mathit{1}}}{r_{\mathit{2}}}.......\left(2\right) \end{gathered}$$

From (1) and (2), we get,:

$\frac{c_1}{c_2}=\frac{r_1}{r_2}=\frac{n_1}{n_2}$

Hence, proved.