A car of mass $m = 1000 k g$ kg is moving with constant speed ν=10m/s on a parabolic shaped bridge AFOE of span $l = 40 m$ and height $h = 20 m$ as shown in the figure ( $g = 10 m / s^{2}$ ) . Then the net force applied by the bridge on the car when the car is at point F, is –

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$\frac{5000}{\sqrt{2}} N$

$5000 \sqrt{\frac{2}{5}} N$

$\frac{10000}{\sqrt{2}} N$

$5000 \sqrt{\frac{5}{2}} N$

answer is A.

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Detailed Solution

If ‘c’ is the origin & co is Y-axis AE is the X-axis, then equation of the parabola is
$\begin{array}{l} y = - A x^{2} + B \\ W h e r e B = h = 20 \\ A = \frac{B}{{(\frac{l}{2})}^{2}} = \frac{1}{20} \\ y = 20 - \frac{x^{2}}{20} \end{array}$
$\begin{array}{l} \frac{d y}{d x} = - \frac{x}{10} = 1 a t x = - 10 i . e a t F \\ \frac{d^{2} y}{d x^{2}} = \frac{- 1}{10}, ∴ R = 20 \sqrt{2} a t F \end{array}$
$m g \cos θ - N = \frac{m v^{2}}{R}$ where $θ =$ angle made by tangent at F $= 45^{0}$
$\begin{array}{l} \frac{10^{4}}{\sqrt{2}} - N = \frac{10^{4}}{2 \sqrt{2}} \\ N = \frac{10^{4}}{2 \sqrt{2}} & f r i c t i o n = m g \sin θ = \frac{10^{4}}{\sqrt{2}} \\ N e t c o n t a c t f o r c e = 5000 \sqrt{\frac{5}{2}} \end{array}$