question_answer

P is the point of contact of a wheel and the ground. The radius of wheel is 1 m. The wheel rolls on the ground without slipping. The displacement of point P when wheel completes half rotation is:                         [AIPMT 2002]

A) 2 m

B) \[\sqrt{{{\pi }^{2}}+4}\,m\]

C) \[\pi \,m\]

D) \[\sqrt{{{\pi }^{2}}+2}\,m\]

Correct Answer: B

Solution :

When the wheel rolls on the ground without slipping and completes half rotation, point P takes new position as P’ as shown in figure. Horizontal displacement, \[x=\pi R\]

Vertical displacement, \[y~=2R\]

Thus, displacement of the point P when wheel completes half rotation,

\[s=\sqrt{{{x}^{2}}+{{y}^{2}}}\]

\[=\sqrt{{{(\pi R)}^{2}}+{{(2R)}^{2}}}\]

\[=\sqrt{{{\pi }^{2}}{{R}^{2}}+4{{R}^{2}}}\]

But       \[R=1\text{ }m\] (given)

\[\therefore \]      \[s=\sqrt{{{\pi }^{2}}{{(1)}^{2}}+4{{(1)}^{2}}}\]

\[=\sqrt{{{\pi }^{2}}+4}\,m\]