Question

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

Answer 1

Hint Concept of limiting friction \[\left( \mu =\tan \theta \right)\] comes into picture. By calculating slope \[\left( \dfrac{dy}{dx} \right)\] and then comparing with the above equation, the value of \[x\] and hence, \[y\] can be calculated.
Formula used: \[\mu =\tan \theta \] is the equation of limiting friction.

Step by step solution
Given that,
$\Rightarrow$ \[\text{y = }\dfrac{{{x}^{3}}}{6}\] …..(1)
We know that, slope is given by:
Slope \[=\dfrac{dy}{dx}\]
Therefore, differentiating (1) with respect to \[x\], we get
\[\begin{align}
  & \dfrac{dy}{dx}=\dfrac{d}{dx}\left[ \dfrac{{{x}^{3}}}{6} \right] \\
 & =\dfrac{1}{6}.3{{x}^{2}}
\end{align}\]
Slope\[=\dfrac{{{x}^{2}}}{2}\] …..(2)
Now, in this question, the block is under limiting friction. Limiting friction is the maximum value of static friction that comes into play when a body is just at a point of sliding over.
In case of limiting friction, we have
\[\tan \theta =\mu \] …..(3)
We know that, \[\tan \theta \] gives the value of slope. Hence, the left hand side of equations (2) and (3) are the same. Therefore, comparing right hand sides, we get
$\Rightarrow$ \[\mu =\dfrac{{{x}^{2}}}{2}\]
$\Rightarrow$ \[\mu =0.5\], given
$\Rightarrow$ \[\therefore \dfrac{{{x}^{2}}}{2}=0.5\]
\[\begin{align}
  & {{x}^{2}}=0.5\times 2 \\
 & {{x}^{2}}=1 \\
 & x=1\text{ m}
\end{align}\]
If \[x=1\], then putting this value in equation (1), we get
$\Rightarrow$ \[\text{y = }\dfrac{1}{6}\text{m}\]

Correct answer is (A).

Note
Coefficient of friction is a value that shows the relationship between two objects and the normal reaction between the objects that are involved.
This coefficient can be of two two types : coefficient of static friction and coefficient of dynamic friction. The coefficient of static friction is the frictional force between two objects when neither of the objects is moving.
Coefficient of dynamic friction is the force between two objects when one is moving or if two objects are moving against one another.
now the laws of limiting friction are as follows:
1. The direction of limiting frictional force is always opposite to the direction of motion.
2. Limiting friction acts tangential to the two surfaces in contact.
3. The magnitude of limiting friction is directly proportional to the normal reaction between the two surfaces.