Question

A parabolic bowl with its bottom at origin has the shape $y=\frac{x^2}{20}$ . Here $x$ and $y$ are in metres. The maximum height at which a small mass $m$ can be placed on the bowl without slipping (coefficient of static friction is $0.5$ ) is

• A. $2.5m$
• B. $1.25m$
• C. $1.0m$
• D. $4.0m$

Answer 1

Answer: (B)

Slope, $\frac{dy}{dx}=\frac{x}{10}$

or $tan\theta =\frac{x}{10}$ ... (i)
Equilibrium of mass in horizontal direction gives the equation,
$\mu Ncos\theta =Nsin\theta$
or $tan\theta =\mu =\frac{1}{2}$ ... (ii)
From Equations. (i) and (ii)
$\frac{x}{10}=\frac{1}{2}$
or $x=5m$
$\therefore$ $y=\frac{x^2}{20}=\frac{25}{20}=1.25m$