Question

A point $P$ (on a wheel) is the contact point of the wheel and the ground. The wheel rolls on the ground without slipping. The value of displacement of the point $P$, when the wheel completes half a rotation is (if radius of 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 $\sqrt{{\pi }^2+4}\text{m}$


Given, the wheel rolls without slipping. Hence, the distance covered by the point $P$ in horizontal direction will be equal to the half of the perimeter of the circle i.e $\frac{2\pi R}{2}=\pi R$
In half rotation, point $P$ will be at the top as shown in figure.
Thus, displacement of point $P$ (by Pythagoras theorem)
$r=\sqrt{(\pi R{)}^2+(2R{)}^2}$
$r=R\sqrt{{\pi }^2+4}$
$(\because R=1\text{m})$
$r=\sqrt{{\pi }^2+4}\text{m}$