Question

The figure shows an accelerometer, a device for measuring the horizontal acceleration of cars and airplanes. A ball is free to roll on a parabolic track described by the equation $y=x^2$, where both $x$ and $y$ are in meters. A scale along the bottom is used to measure the ball's horizontal position $x$. The mass of ball is $\frac{4}{70}\text{kg}$. The accelerometer is free to move along the horizontal.

What is the magnitude of normal reaction (in $\text{N}$) by track on ball if ball remains at rest at $x=2\sqrt{3}$.
(Given that, $\sin \theta =\frac{4\sqrt{3}}{7}$), where $\theta$ is the angle made by the tangent with the horizontal at this point. Take $g=10m{s}^{-2}$

Answer 1

From FBD,
$N\cos \theta =mg$
and $N\sin \theta =ma$
$\Rightarrow \tan \theta =\frac{a}{g}$
Now, $y=x^2$
$\Rightarrow y^{\prime }=2x=4\sqrt{3}$ (Since $x=2\sqrt{3}$)


$\tan \theta =y^{\prime }=2x$
$\Rightarrow 2x=\frac{a}{g}$
$\Rightarrow a=2xg$
Again,
$N=\frac{ma}{\sin \theta }=\frac{2mgx}{\sin \theta }(\because \sin \theta =\frac{4\sqrt{3}}{7})$
$\Rightarrow N=\frac{2\times \frac{4}{70}\times 10\times 2\sqrt{3}}{\frac{4\sqrt{3}}{7}}=4\text{N}$