Question

How a single force is resolved along the perpendicular axis acting an angle theta with horizontal?​

Answer 1

Answer:

A single force, which is acting at angle θ from a horizontal axis, can be resolved into components which act along the perpendicular axis.

Consider the perpendicular axis x and y, where x represents the horizontal axis and y represents vertical axis.

The Force is resolved into 2 parts, one acts along x-axis and is represent by X. The other acts along y-axis and is represented by Y.

From the diagram we can see that the Force and its components X and Y makes up a right angles triangle, where θ is the angle from the x-axis

Find X:

We know that:

cosθ = Base/Hypotenuse

cosθ = X/F

X = Fcosθ

Find Y:

We know that:

sinθ = Perpendicular/Hypotenuse

sinθ = Y/F

Y = Fsinθ

Relation of Force and its Components:

Force F can be represent by:

F = Fcosθ (along x-axis) + Fsinθ (along y-axis)

As they form a right angled triangle, we can use Pythagoras Theorem to show the relation between Force and its components.

Hypotenuse² = Base² + Perpendicular²

F² = X² + Y²

F² = (Fcosθ)² + (Fsinθ)²

$F = \sqrt{(F\cos\theta)^2+(Fsin\theta)^2}$

Where θ can be found by using any of the trignometric functions.