Question

A parabolic bowl with it's bottom at origin has the shape $y=\frac{x^2}{20}$. Here $x$ and $y$ are in meters. 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. 2.5 m
• B. 1.25 m
• C. 1.0 m
• D. 4.0 m

Answer 1

The correct option is B 1.25 m

$y=\frac{x^2}{20}$

$\therefore \frac{dy}{dx}=\frac{x}{10}=tan\theta$ .....(1)

in Horizontal direction $\mu Ncos\theta =Nsin\theta$

$\tan \theta =\mu =\frac{1}{2}$ ....(2)

from equation (1) and equation (2)

$\frac{x}{10}=\frac{1}{2}$ or $x=5m$

$\therefore y=\frac{x^2}{20}=\frac{25}{20}=\frac{5}{4}=1.25m$