Question

A wheel of radius $3m$ rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is:

• A. $2\pi m$
• B. $\sqrt{2\pi }m$
• C. $\sqrt{{\pi }^2+4}m$
• D. $3\sqrt{{\pi }^2+4}$

Answer 1

Answer: (D)

Horizontal distance covered by the wheel in half revolution $=\pi \mathrm{R}=3\pi$

Net displacement of the point which was initially in contact with ground.

\begin{aligned}
\mathrm{R}=\mathrm{AA}^{\prime}=\sqrt{(\pi \mathrm{R})^{2}+(2 \mathrm{R})^{2}} & \\
\begin{aligned}
=\sqrt{(3 \pi)^{2}+(2 \times 3)^{2}} &=\sqrt{9 \pi^{2}+36} \\
&=3 \sqrt{\pi^{2}+4}
\end{aligned}
\end{aligned}