Question

A car of mass $m=1000\text{kg}$ is moving with constant speed $v=10\text{m/s}$ on a parabolic shaped bridge $AFOE$ of span $l=40\text{m}$ and height $h=20\text{m}$ as shown in the figure. Then the net force applied by the bridge on the car, when it is at point $F$ is :

Assume $g=10{\text{ms}}^{-2}$

• A. $5000\sqrt{\frac{5}{2}}\text{N}$
• B. $\frac{5000}{\sqrt{2}}\text{N}$
• C. $\frac{10000}{\sqrt{2}}\text{N}$
• D. $5000\sqrt{\frac{2}{5}}\text{N}$

Answer 1

The correct option is A $5000\sqrt{\frac{5}{2}}\text{N}$

Taking the origin at $O$ and $OC$ as y-axis equation of bridge is
$y=\frac{x^2}{20}$
So, slope at $F=\tan \theta =\frac{dy}{dx}=\frac{2x}{20}=\frac{2\times 10}{20}=1$
At point $F$,
$mg\cos \theta -N=\frac{m{v}^2}{R}$
$mg\frac{1}{\sqrt{2}}-N=\frac{m{v}^2}{R}$

$N=\frac{mg}{\sqrt{2}}-\frac{m{v}^2}{R}$ and $f_s=mg\sin \theta =\frac{mg}{\sqrt{2}}$
Radius of curvature at $F$, $R=\frac{{(1+{\left(\frac{dy}{dx}\right)}^2)}^{\frac{3}{2}}}{\left|\frac{d^{2}y}{d{x}^2}\right|}=\frac{(1+1{)}^{3/2}}{\frac{1}{10}}=20\sqrt{2}\left(\text{metre}\right)$

$N=\frac{10m}{\sqrt{2}}-\frac{100m}{20\sqrt{2}}=\frac{5m}{\sqrt{2}}$
$f_s=\frac{10m}{\sqrt{2}}$
$\Rightarrow \text{Net force}=m\sqrt{\frac{25}{2}+\frac{100}{2}}=5m\sqrt{\frac{5}{2}}=5000\sqrt{\frac{5}{2}}N$