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:

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