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A wooden block is placed on an inclined plane. The block just begins to slide down when the angle of the inclination is increased to $45°$. What is the increased coefficient of the friction?
A) 0.25
B) 0.75
C) 1
D) 0.5

Last updated date: 20th Jun 2024
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Hint: Frictional force is the force generated by two surfaces that comes in contacts with each other and slides against each other.
$f = \mu N$ (f is the force of friction, $\mu $ is coefficient of friction, N is the normal force)
Coefficient is given as;
$\mu = \tan \theta $ ($\theta $ is the angle of inclination)
Using the above two relations we will calculate the coefficient of friction when angle of inclination is given.

Complete step by step solution:
Let us first Define Force of Friction for better understanding of the question asked:
Friction is the force resisting the relative motion of solid surfaces, fluids layers and material elements sliding against each other. Frictions are of many types such as: Dry friction, Fluid friction, Lubricating friction, skin friction, Internal friction. When surfaces in contact slides across each other then friction between the two surfaces converts kinetic energy into thermal energy.
Now we will calculate the coefficient of friction:
A body when about to slide experiences a normal for perpendicular to its surface, force of gravity acts downward and frictional force which resists the movement of the wooden block.
Weight of the body (gravitational force) mg can be resolved into two components which are:
$mg\cos \theta $ and $mg\sin \theta $, the component which is action at the base of the wooden block or the surface which is touching the inclined plane will have force of friction acting on it.
As per the Equilibrium:
Both the components equalize each other
$\mu mg\cos \theta = mg\sin \theta $ ($\mu $ is the coefficient of friction)
On substituting the value angle we have the equation;
$ \Rightarrow \mu mg\cos 45° = mg\sin 45°$( On cancelling the common terms)
$ \Rightarrow \mu = \dfrac{{\sin 45°}}{{\cos 45°}}$
$ \Rightarrow \mu = \tan 45° $
$\Rightarrow \mu = 1 $ (tan45° is 1)

Thus, option (C) is correct.

Note: Force of friction is the most important force which has many applications in our daily life such as, the marks below our shoes and slippers to resistance them from sliding, marks made on the tyres of the vehicles even bicycle to prevent the vehicles from sliding, resistance offered by rotation of the rotor of the motors for proper rotation.