### Chapter : Newton's Laws Of Motion & Friction

__9. Motion of a body on a smooth inclined plane__

__9. Motion of a body on a smooth inclined plane__

**9. Motion of a body on a smooth inclined plane**

A body is placed on a smooth inclined plane AB which makes an angle q with the horizontal. The forces acting on body are

**(i)**Weight of the body mg acting vertically down.

**(ii)**Normal reaction R acting perpendicular to the plane.

The weight mg of the body is resolved parallel and perpendicular to the plane as mg sin θ parallel to the plane and mg cos θ perpendicular to the plane.

Thus ma = mg sin θ ⇒ a = g sin θ ...(i)

R = mg cos θ ...(ii)

**Note:**

The same result can also be obtained by resolving the forces horizontally and vertically.

R sin θ = ma cos θ

mg – R cos θ = ma sin θ

solving we get,

a = g sin θ, R = mg cos θ

**Special case :**

When the smooth plane is moving horizontally with an acceleration (b) as shown in fig

In this case :

m (a + b cos θ) = mg sin θ

and mb sin θ = R – mg cos θ

solving we get a = g sin θ – b cos θ

R = m (g cos θ + b sin θ)

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