Question

A wheel of radius $R$ is trapped in a mud pit and spinning. As the wheel is spinning, it splashes mud blobs with initial speed $u$ from various points on its circumference. The maximum height from the centre of the wheel, to which a mud blob can reach is

• A. $u^2/2g$
• B. $\frac{u^2}{2g}+\frac{g{R}^2}{2u^2}$
• C. $0$
• D. $R+\frac{u^2}{2g}$

Answer 1

Answer: (B)

Let a mud blob is detached from the circumference of wheel with initial speed $u$ at angle $\theta$ as shown below.

Height upto which mud blob can be thrown is
$h=$ Maximum height of projectile $+$ Height at which mud blob thrown is
$\therefore h=\frac{u^2{\sin }^2\theta }{2g}+R\cos \theta \dots \left(i\right)$
$h$ is maximum when $\frac{dh}{d\theta }=0$
$\Rightarrow \frac{d}{d\theta }\left(\frac{u^2{\sin }^2\theta }{2g}+R\cos \theta \right)$
$\Rightarrow \frac{u^2}{2g}\cdot 2\sin \theta \cos \theta -R\sin \theta =0$
$\Rightarrow \frac{u^2}{g}\cos \theta =R$ or $\cos \theta =\frac{Rg}{u^2}$
$\Rightarrow {\sin }^2\theta =1-\frac{R^2{g}^2}{u^2}$
Substituting these values in Eq $\left(i\right),$ we get
${\lambda }_{\max }=\frac{u^2}{2g}\left(1-\frac{R^2{g}^2}{u^4}\right)+R\left(\frac{Rg}{u^2}\right)$
$=\frac{u^2}{2g}-\frac{R^{2}g}{2u^2}+\frac{R^{2}g}{u^2}=\frac{u^2}{2g}+\frac{R^{2}g}{2u^2}$