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A parabolic bowl with its bottom at origin has the shape $y = \dfrac{{{x^2}}}{{20}}$. 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) $2.5m$
B) $1.25m$
C) $1.0m$
D) $4.0m$

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Last updated date: 23rd May 2024
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Answer
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Hint: In order to solve the given question, first of all we need to find the value of $x$. After that we can put the value of $x$ in the equation for maximum height and get the required value. Now, for finding the value of $x$ we need to find the slope of the curve and we need to find the forces acting in the horizontal direction. Then we can finally conclude with the correct solution of the given question.

Complete step by step solution:
The shape of the parabolic bowl is given as, $y = \dfrac{{{x^2}}}{{20}}$
Now, let us find the slope of the bowl. So, the equation can be written as,
$\Rightarrow \tan \theta = \dfrac{{dy}}{{dx}}$
$ \Rightarrow \tan \theta = \dfrac{{d({x^2})}}{{dx(20)}} = \dfrac{{2x}}{{20}} = \dfrac{x}{{10}}$…………… (i)
We know that in the horizontal direction,
$\Rightarrow \mu N\cos \theta = N\sin \theta $
$ \Rightarrow \tan \theta = \mu $……………….. (ii)
The value of coefficient of friction is given as,$\mu = 0.5$
Now, from equation (i) and (ii), we can write it as,
$\Rightarrow \dfrac{x}{{10}} = 0.5$
$\therefore x = 5m$
Now, we need to find the maximum height. So, from the figure, for maximum height we need to take $y = \dfrac{{{x^2}}}{{20}}$.
After putting the value of $x$ in the above equation, we get,
$\Rightarrow y = \dfrac{{{x^2}}}{{20}} = \dfrac{{25}}{{20}} = \dfrac{5}{4} = 1.25m$
Therefore, the value of maximum height is $1.25m$.

Hence, we can conclude that option (B), i.e. $1.25m$ is the correct choice of the given question.

Note: From Newton’s third law of motion, we know that every action force has an equal reaction force which acts in the opposite direction. In case when the body is in a static position, the forces acting on the body have to be balanced or in other words we can say that their sum has to be zero.