Turning Effect of Forces

Like Parallel Forces	Unlike Parallel Forces
Forces that act in the same direction and have equal magnitudes.	Forces that act in opposite directions and have unequal magnitudes.
This results in a net force equal to the sum of the forces.	This results in a net force not equal to the sum of the forces.
Two people pushing a couch in the same direction.	Two people pushing a couch in opposite directions with different strengths.

Modern communication relies on artificial satellites orbiting the Earth in geostationary orbits, facilitated by the principles of force and its turning effects. Operating vehicles and balancing on bicycles or tightropes also requires an understanding of these forces and their impact.

A ceiling fan is suspended in a hook through a supporting rod. The forces acting on it are;

Weight of the fan acting vertically downwards.
Tension in the supporting rod pulls it vertically upwards.

Addition of Forces:

The resulting force that has the same effect as the combination of individual forces is known as the resultant force. Adding forces cannot be performed using standard arithmetic rules. Instead, two methods are used to add forces, which are generally used to add vectors:

Graphical Method.
Analytical Method.

Graphical Method: This method is used for the addition of One-dimensional vector quantities through the head-to-tail rule method.

Addition Through Head To Tail Rule:

Draw a diagram with the tail of the first vector at the origin and the head of the first vector at the initial point.
Place the tail of the second vector at the head of the first vector, so that the two vectors are end-to-end.
Draw the vector that starts at the tail of the first vector and ends at the head of the second vector. This is the sum or resultant vector.
Label the magnitude and direction of the resultant vector.
Repeat this process for adding more vectors if necessary.

Trigonometric Ratios:

In right-angled triangles, specific names are assigned to the ratios of any two sides. There are a total of six ratios, including three main ratios, namely sine, cosine, and tangent, which are widely used in physics. The other three ratios are their reciprocals.

RESOLUTION OF FORCES

The process of splitting a vector into mutually perpendicular components is called the resolution of vectors.

Consider a triangle ΔOAB such that AB is perpendicular on OB

OB= Fx (as it is parallel to the x-axis)

BA=Fy (as it is parallel to the y-axis)

Both are perpendicular to each other.

Line OA is the resultant vector called F.

Therefore, the magnitude of the resultant vector can be given as,

F = Fx Fy

The trigonometric ratios can be used to find the magnitude of Fx and Fy in the right-angled

triangle ΔOAB.

For x component

cos θ= \(\frac{base}{hypotenuse}\)= \(\frac{OB}{OA}\)= \(\frac{F_x}{F}\)

cos θ= \(\frac{F_x}{F}\)

Fx= Fcosθ

For y component

sinθ=\(\frac{perpendicular}{hypotenuse}\)= \(\frac{AB}{OA}\)= \(\frac{F_y}{F}\)

sinθ= \(\frac{F_y}{F}\)

\({F_y}\)= Fsinθ

Resultant Vector:

Suppose Fκ and Fy are the perpendicular components of x and y, the force Fκ and

Fy are represented by line segments AB and OB. According to head to tail rule,

OA represents the resultant vector which is F Thus,

F = Fx + Fγ

In order to find the magnitude of F. Apply the Pythagorean Theorem to right angled

triangle \(ΔOAB\).

According to the Pythagorean Theorem

\(hyp^2=perp^2+base^2\)

\(F^2=Fy^2+Fx^2\)

To find the resultant F take the square root on both sides

\(\sqrt{F}= \sqrt{F_x^2 + F_y^2}\)

\(F=\sqrt{F_x^2 + F_y^2}\)

The direction of results can be obtained as

\(Tan∅= \frac{base}{hypotenuse}=\frac{F_x}{F_y}\)

\(∅= Tan^-1(\frac{F_x}{F_y})\)