Fewpal
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Write different equation of motion on inclined plane for the following cases. 

(a) Motion on inclined plane when surface is smooth. 

(b) Motion on inclined plane when surface is rough.

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 CASE: 1 WHEN SURFACE SMOOTH

Fig(a) shows a body of weight W, sliding down on a smooth inclined plane. 

Let,

θ = Angle made by inclined plane with horizontal

a = Acceleration of the body

m = Mass of the body = W/g

Since surface is smooth i.e. frictional force is zero. Hence the force acting on the body are its own weight W and reaction R of the plane. 

The resolved part of W perpendicular to the plane is Wcos θ, which is balanced by R, while the resolved part parallel to the plane is Wsin θ, which produced the acceleration down the plane. Net force acting on the body down the plane.

F = W.sin θ, but F = m.a

m.a = m.g.sinθ

i.e. a = g.sin θ (For body move down due to self weight.)

and, a = -g.sin θ (For body move up due to some external force) 

CASE: 2 WHEN ROUGH SURFACE 

Fig(b) shows a body of weight W, sliding down on a rough inclined plane. 

Let,

θ = Angle made by inclined plane with horizontal 

a = Acceleration of the body 

m = Mass of the body = W/g 

µ = Co-efficient of friction 

Fr = Force of friction

when body tends to move down:

R = w.cosθ

Fr = µ.R = µ.W.cosθ

Net force acting on the body F = W.sinθ - µ.W.cosθ

i.e. m.a = W.sinθ - µ.W.cosθ

 Put m = W/g we get

a = g.[sinθ - µ.cosθ] (when body tends to move down)

a = –g.[sinθ - µ.cosθ] (when body tends to move up)

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