Question

A wedge of mass 2m and a cube of mass m are shown in the figure. Between the cube and wedge, there is no friction. The minimum coefficient of friction between the wedge and the ground so that the wedge doesn’t move is
A. $0.2$
B. $0.25$
C. $0.10$
D. $0.50$

Answer 1

When the object is in motion, many forces are acting on it. One of the forces is the frictional force. The frictional force is directly proportional to the normal force acting the body.

Formula used:
$\lambda \sin \theta = {F_r}$
Where, ${F_r}$=Frictional force

Complete step by step answer:
We know that the coefficient of friction is the ratio of the normal force to the frictional force. The coefficient of friction is represented as $\mu$. The frictional force is divided into horizontal and vertical components. Those are $\lambda \sin \theta$ and $\lambda \cos \theta$.
$\Rightarrow \lambda \sin \theta = {F_r}$
$\Rightarrow \lambda \sin \theta = \mu \left( {2mg + mg{{\cos }^2}\theta } \right)$
$\Rightarrow mg\sin \theta \cos \theta = \mu \left( {2mg + mg{{\cos }^2}\theta } \right)$
$\Rightarrow \dfrac{1}{2} = \mu \left( {2 + \dfrac{1}{2}} \right)$
$\Rightarrow \dfrac{1}{2} = \mu \left( {\dfrac{5}{2}} \right)$
$\Rightarrow \mu = \dfrac{1}{5}$
$\therefore \mu = 0.5$

Hence the correct option is option D.

Note:
(i) The coefficient of friction is the dimensionless number that is defined as the ratio between the normal force to the frictional force.
(ii) Materials having a coefficient of friction smaller than 0.1 are called lubricous materials which means it has very low friction. The coefficient of friction of materials depends on the nature of the materials and the surface roughness.
(iii) The coefficient of friction, $\mu$, is a measure of friction between the existing two surfaces. Between the solid surfaces, fluid layers, and material elements sliding against each other, the frictional force acts as a resistive force.
(iv) There are many instruments used to measure friction. Those are tribometers and profile graphs. Friction has its application in many areas from the brakes in automobiles to the matchsticks in our kitchen.
(v) There are four types of frictions. They are static friction, rolling friction, sliding friction, and fluid friction. The static, sliding and rolling frictions happen between the solid surfaces. Whereas fluid friction occurs in fluids.