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
\[\tan \beta =\frac{BC}{AB}=\left[ \frac{2R}{\pi R}
\right]=\left[ \frac{2}{\pi } \right]\]
\[\therefore \] \[\beta ={{\tan }^{-1}}\left(
\frac{2}{\pi } \right)\]
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