PROVE THAT THE ANGLE OF REPOSE IS EQUAL TO THE ANGLE OF FRICTION I.E. α = φ

From the condition of equilibrium

(  ͢+) Σf_x = 0

F – Wsinα = 0

F = Wsinα …………….(1)

(+↑)Σf_y = 0

R – cosα = 0

R = cosα ………………….(2)

Again F = μR

From equation (1)

F = μR = Wsinα

μ x Wcosα = Wsinα

μ = Wsinα/ Wcosα = tanα = tanφ

or α = φ

Alternatively

tanφ F/R = Wsinα/ Wcosα = tanα

Or φ = α