Question

A point $P$ is the contact point of wheel on the ground which rolls on ground without slipping. Find the value of displacement of the point $P$ when wheel completes half revolution. Radius of the wheel is $1\text{m}$.

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

Answer 1

The correct option is B $\sqrt{{\pi }^2+4}\text{m}$

When a body is under pure rolling then the moment it completes $1$ revolution, the horizontal distance covered is equal to perimeter of circle i.e $2\pi r$
given, radius of wheel is $r=1\text{m}$


Initial position, $P$ is in contact with ground
After $\frac{1}{2}$ revolution it reached the highest point $B$ on wheel.
$\Rightarrow$ Horizontal distance $PA$ given by,
$x=\frac{2\pi r}{2}=\pi r=\pi \times 1$
From the right angled triangle $\Delta PBA$, applying pythagoras theorem:
$PB=\sqrt{P{A}^2+A{B}^2}=\sqrt{(\pi \times 1{)}^2+(2\times 1{)}^2}$
$\therefore$Displacement of point $P$ ,
$PB=\sqrt{{\pi }^2+4}\text{m}$