# The Inclined Plane

by • 04/07/2012 • GeneralComments (0)570

## The Inclined Plane

A plane which makes certain angle θ with the horizontal is called an inclined plane.

Diagram Coming Soon

Consider a block of mass ‘m’ placed on an inclined plane making certain angle θ with the horizontal. The forces acting on the block are

1. W, weight of the block acting vertically downward.

2. R, reaction of the plane acting perpendicular to the plane

3. f, force of friction which opposes the motion of the block which is moving downward.

Diagram Coming Soon

Now we take x-axis along the plane and y-axis perpendicular to the plane. We resolve W into its rectangular components.

Therefore,

Component of W along x-axis = W sin θ

And

Component of W along y-axis = W cos θ
1. If the Block is at Rest

According to the first condition of equilibrium

Σ Fx = 0

Therefore,

f – W sin θ = 0

Or

f = W sin θ

Also,

Σ Fy = 0

Therefore,

R – W cos θ = 0

Or

R = W cos θ
2. If the Block Slides Down the Inclined Plane with an Acceleration

Therefore,

W sin θ > f

Net force = F = W sin θ – f

Since F = m a and W = m g

Therefore,

m a = m g sin θ – f
3. When force of Friction is Negligible

Then f ≈ 0

Therefore,

equation (3) => m a = m g sin ≈ – 0

=> m a = m g sin ≈

or a = g sin ≈ …………. (4)
Particular Cases

Case A : If the Smooth Plane is Horizontal Then 0 = 0º

Therefore,

Equation (4) => a = g sin 0º

=> a = g x 0

=> a = 0
Case B : If the Smooth Plane is Vertical Then θ = 90º

Therefore,

Equation (4) => a = g sin 90º

=> a = g x 1

=> a = g

This is the case of a freely falling body.

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