question_answer

What is the displacement of the point of a wheel of radius R initially in contact with the ground when the wheel rolls forward half a revolution? Let x axis be the forward direction.

Answer:

Refer to Fig. 2(HT).4., when wheel completes half revolution, then point A on wheel reaches at C. The horizontal distance covered = \[AB=\pi R\]; while vertical distance covered is = BC = 2R.                 Displacement of point A on the wheel is =\[\overset{\to }{\mathop{AC}}\,\]             Here, \[AC=\sqrt{A{{B}^{2}}+B{{C}^{2}}}=\sqrt{{{\left( \pi R \right)}^{2}}+{{\left( 2R \right)}^{2}}}\]                                                 = \[R\sqrt{{{\pi }^{2}}+4}\] If \[\beta \] is the angel which\[\overset{\to }{\mathop{AC}}\,\] makes with \[\overset{\to }{\mathop{AB}}\,\], then \[\tan \beta =\frac{BC}{AB}=\left[ \frac{2R}{\pi R} \right]=\left[ \frac{2}{\pi } \right]\] \[\therefore \]   \[\beta ={{\tan }^{-1}}\left( \frac{2}{\pi } \right)\]