Angle of repose

 Angle of repose  ( θo )

It is the maximum angle of rough incline at which block  remain at rest  on the incline surface . 

It is represented by θo . 

Formula ,   θo = Tan-1 (μ)

Along perpendicular to incline 

fnet = 0 

N - mgcosθ =0

N = mgcosθ

frl = μN

frl =  μmgcosθ

Along incline 

Fnet = ma

mgsinθ - frs = m*0

mgsinθ - frs = 0 

mgsinθ  =  frs 

As θ increases the value of sinӨ increases , so tendency of motion is increases . 

body will remain at rest if 

frl >= mgsinθ

At limiting equilibrium 

frl = mgsinθ

μmgsinθ =  mgsinθ 

Tanθo = μ

θo = Tan-1μ , here  θo is the angle of repose 

if the angle is less than angle of repose then the force of friction is static . 

θ < θo   frs = mgsinθ       here frs is a static friction 

if the angle is equal to angle to repose , then force of friction is limiting 

θ = θo , frL = μsmgcosθ  here frL is a static limiting friction  , μs is a coefficient of static friction .

If the angle is greater than angle of repose body will not remain at rest on the incline surface , then force of friction will act is kinetic . 

θ > θo  frk = μkmgcosθ   here frk is a kinetic friction  , μk is coefficient  of kinetic friction.

How  to find the acceleration of the body which is moving downward on a rough  incline surface ?  

here θ is greater than θo ; body will not remain at rest on the incline surface . so the body will move downward to the incline surface . In this case kinetic friction force will act in the opposite direction motion of body . 

Along perpendicular to the incline 

fnet = o 

N - mgcosθ = 0 

N = mgcosθ    equation (1) 

Along incline 

 fnet = ma 

mgsinθ - frk = ma 

mgsinӨ - μkmgcosθ = ma   ( frk = μN = μ .mgcosθ  ; from equation 1  ) 

gsinӨ - μkcosθ = a 

hence no any body remain at rest on the incline surface  , for θ is greater than θo . 

so the body will  move  at an acceleration of   gsinӨ - μkcosθ  in the downward direction of  the rough incline surface .