Question

A block of mass m is placed on a surface with a vertical cross section given by $y = \dfrac{{{x^3}}}{6}$, if the coefficient of friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is:
$\left( A \right)\dfrac{1}{2}$m
$\left( B \right)\dfrac{1}{6}$m
$\left( C \right)\dfrac{2}{3}$m
$\left( D \right)\dfrac{1}{3}$m

Answer 1

Hint: In this question use the concept that the block is under limiting friction so the slope of the surface at that point of time is equal to the coefficient of friction so use this property to reach the solution of the question.
Formula used – $\mu = \dfrac{{dy}}{{dx}} = \tan \theta $

Complete Step-by-Step solution:
We have to place the block at the maximum height so that it is not slipping.
So the block is under limiting friction (i.e. if we move the block further by an inch it will slip).
So the limiting friction is equal to the slope of the surface at that point of time.
Now coefficient of friction is often denoted by $\left( \mu \right)$
And slope is nothing but $\dfrac{{dy}}{{dx}} = \tan \theta $
$ \Rightarrow \mu = \dfrac{{dy}}{{dx}} = \tan \theta $................... (1)
Now the equation of the surface is given as
$y = \dfrac{{{x^3}}}{6}$
Now differentiate it w.r.t x we have,
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{{{x^3}}}{6}} \right)$
Now as we know $\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}$ so use this property in above equation we have,
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{6}.3{x^2}$
Now from equation (1) we have,
$ \Rightarrow \mu = \dfrac{{3{x^2}}}{6} = \dfrac{{{x^2}}}{2}$
Now it is given that coefficient of friction is = 0.5
$ \Rightarrow 0.5 = \dfrac{{{x^2}}}{2}$
$ \Rightarrow {x^2} = 1$
Now take square root on both sides we have,
$ \Rightarrow x = \sqrt 1 = 1$
Now substitute this value in the equation of surface we have,
 $ \Rightarrow y = \dfrac{{{x^3}}}{6} = \dfrac{1}{6}$
So the maximum height above the ground at which the block can be placed without slipping is (1/6) m.
So this is the required answer.
Hence option (B) is the correct answer.

Note – Whenever we face such types of questions always recall the property of limiting friction which is stated above and the slope of any surface is the differentiation of that function w.r.t. the given variable so just equate them as above and simplify we will get the required maximum height above the ground at which the block can be placed without slipping.