Question

Angle of friction and angle of repose are ___________?

Answer 1

Hint:The relationship between angle of repose and angle of friction can be easily obtained using their respective formulae and equating them. The numerical or mathematical formula for angle of repose and angle of friction are equated and then their respective formulae are used to obtain the relationship between angle of repose and angle of friction.

Formula Used:The expression for angle of friction is given as:
$\tan \alpha = \mu $ Here $\alpha $ is the angle of friction and $\mu $ is the coefficient of limiting friction.
The expression for angle of repose is given as:

$\tan \theta = \mu $ Here $\beta $ is the angle of repose and $\mu $ is the coefficient of limiting friction.

Complete step by step solution:
We already know that the coefficient of limiting friction is equal to the tangent of the angle of friction. Thus,

$\tan \alpha = \mu $

We also know that the tangent of the angle of repose is numerically equal to the coefficient of limiting friction. Thus we can write:

$\tan \theta = \mu $

Since both tangent of the angle of friction and the tangent of the angle of repose are numerically equal to each other, we can say that they are both numerically equal to each other. That is:

$\tan \theta = \tan \alpha $

Since the tangents of both angle of friction and angle of repose are equal to each other, we can say that they are both numerically equal. So,

$\theta = \alpha $

Thus it is proved that angle of friction and angle of repose are equal.

Additional information:Angle of friction of a body is defined as the angle which is made between the resultant of normal reaction and the direction of force of friction or frictional force.
The angle of repose is defined as that particular angle of an inclined plane at which a body place just begins to slide.

Note:It should be noted that angle of repose and angle of friction are equal only when the object is in an inclined plane as the angle of repose is 0 for an object kept in a horizontal surface. These two equations are crucial in understanding the relationship between angle of repose and angle of friction.