Describe the method of measuring angle of repose

Explanation

Angle of Repose Consider an inclined plane on which an object is placed, as shown in figure. Let the angle which this plane makes with the horizontal be `theta` . For small angles of `theta` , the object may not slide down. As `theta` is increased, for a particular value of `theta` , the object begins to slide down. This value is called angle of repose. Hence, the angle of repose is the angle of inclined plane with the horizontal such that an object placed on it begins to slide.
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Let us consider the various forces in action here. The gravitational force mg is resolved into components parallel `(mg sin theta )` and perpendicular `(mg cos theta ) `to the inclined plane. The component of force parallel to the inclined plane `(mg sin theta)` tries to move the object down. The component of force perpendicular to the inclined plane `(mg cos theta)` s balanced by the Normal force (N).
` N= mg cos theta` .....(1)
When the object just begins to move, the static friction attains its maximum value
`f_s = f_(s)^(max)`
This friction also satisfies the relation
` f_(s)^(max) = mu_s mg sin theta ` .....(2)
Equating the right hand side of equations (1) and (2),
`(f_(s)^(max) )//N = sin theta // cos theta`
From the definition of angle of friction, we also know that
` tan theta = mu_s` .....(3)
in which `theta` is the angle of friction. Thus the angle of repose is the same as angle of friction. But the difference is that the angle of repose refers to inclined surfaces and the angle of friction is applicable to any type of surface.