FORCES ARE VECTORS. This means they have both SIZE and DIRECTION. When adding forces, we must add them like lines, taking LENGTH and ANGLE into account. Adding forces is the same as combining them. When several forces are combined (added) into a single force, this force is called the RESULTANT of those forces.

Lecture Notes Adding-Forces.pdf

Image Video Lesson Description and Link Duration Date Download

Adding Forces in AutoCad (AutoCad old style)

5 min 20091113 6MB
  Adding Forces using AutoCad (AutoCad 2014) 3:23 min 20140217  
  Adding Forces in Excel (Mathematical addition of forces) 3:43 min 20120905 2.8MB

What is a Force?

A "push" or "pull" exerted on a body.
Examples: A person pushing on a wall, Gravity pulling down on a mass, A rope pulling on an object.

Scalars and Vectors

Scalar: Has only a magnitude. Examples: mass, length, time, volume, area, speed, temperature, time.

Vector: Has both a magnitude and direction. Examples: velocity, displacement, or force.

To describe a force we need three things;

  1. Magnitude (Newtons) N
  2. Direction (degrees)
  3. Location
Example: This force has Magnitude = 326N, Direction = 37 degrees, Location = Point A

Converting Angles to 360o Notation

Before we start adding anything, we should define the angle of each force in 360 degree notation. This ensures any mathematical calculations will have the correct signs (e.g. negative X is to the left). It also means the force can be fully defined without using extra diagrams or symbols to help explain direction. Later it comes to calculations later on, 360 degree notation complies with spreadsheet, CAD and other software programmed methods of force analysis.

To convert each force to 360 degree notation, follow these steps;
  1. In the FBD, draw all forces as if they are "pulling". This means the tail of the force connects to the body, and the arrowhead points away from the body.
  2. Take the angle from the positive X direction (horizontal and to-the-right) and go anti-clockwise to the force.


A force has the same effect on a body whether it pushes from one side, or pulls from the other. Consider a train - the loco might pull from the front or push from the back, the effect is the same. So, provided we have the right point of location, and the right direction, we can either push from one side or pull from the other. 

Example: The hammer can be pulled from one side, or pushed from the other. The effect is the same if both the magnitudes (Newtons) and angles (degrees) are equal.


For analysis, we are allowed to move the force ANYWHERE on the line of action of the force. We cannot change the angle or direction, not the point of application.  So PUSH = PULL.


Classes of Forces

There are three classes of force systems that have special methods of analysis.
  1. Colinear. "Same Line" These forces act along the same line. Forces on each end of a tensioned rope are colinear. They all have the same angle, and same location. Because they act in the same line, colinear forces can be added as scalars, e.g. 3+4=7N.
  2. Concurrent. 'Same Point". These forces all go through the same point. Ropes tied to an eyebolt give concurrent forces. These forces can be added as lines with angle. e.g. 3N+4N = 5N (assuming 3N and 4N are perpendicular) 
  3. Coplanar. "Same Plane'. These forces are 2 dimensional. They might have different angles and locations. Addition is complicated because they can create a turning effect around each other.

Mass and Weight

Mass is not the same as weight. Weight is a force and is measured in Newtons. Mass is measured in kilograms, which is really just a count of how many neutrons and protons inside it. So if someone wants to know your weight, give them the answer in Newtons!
Weight is the gravitational force caused by the object's mass. This depends on which planet you are on. It even varies on earth - you would feel lighter at the equator than at the poles, because the spinning earth lifts you up a tad (about 0.2% lighter!).

To find out how much force (weight) a mass will produce,

W = M * g

W = Weight (Newtons)
M = Mass (kg)
g = acceleration due to gravity (9.81m/s2). In Sydney it is closer to 9.8, but the higher figure is most common.

Example: How much does 10kg weigh? No, it's not a trick question.  W = 10 * 9.81 = 98.1N

Other Names for a Force

The following words can also used to describe a force (N);

  • push
  • pull
  • effort
  • tension
  • weight
  • thrust
  • action

These words should only be used for mass (kg);

  • mass

Watch out for this one, sometimes used for force (N) and somtimes mass (kg);

  • load

Note: If unsure, always check the units! Force is always based on Newtons (N)

X & Y Components of a Force

Since our force lies on the XY plane, we can split it into X component (horizontal) and Y component (vertical). Since we are using 360 degree notation for the angle (A), the formula will always give the correct signs. This is a big reason to use 360 notation!

To get components;

Fx = Magnitude * cos (Angle)
Fy = Magnitude * sine (Angle)

Example: Find the X and Y components for a force of 316N at 35 degrees.


Addition of Forces

Parallelogram of Forces

Example: There are two forces (F1 and F2) acting on an object. The same effect could be achieved with only one force which is the sum of F1 and F2. This sum is called the Resultant Force.

This is actually just vector addition. Forces can be treated as lines, and just like drawing a polyline in CAD, you must enter the length (representing Newtons) and the angle (degrees).

Graphical Addition of Forces

In the following example (a) two forces F1 and F2 are applied to the body. (b) Shows these added together in a "force polygon" - just as you would connect lines in a CAD sketch. (c) is the Resultant force R that could replace the original two forces and have exactly the same effect.

In the following example (d) three forces F1, F2 and F3 are applied to the body. (b) Shows these added together in a "force polygon". Notice that the 'polyline' of forces begins at the tail of F1 and head at the arrowhead of F3. (c) The Resultant force R is equal to the original three forces. It was measured from the beginning (tail of F1) to the end (head of F3).

Graphical method instructions: How to add forces in Solid Edge

Mathematical Addition of Forces

Step 1. Resolve into components

To avoid using CAD or triangle geometry (cosine rule etc), we can use the colinear force trick. Colinear forces can be added like numbers! So if every force is resolved into X and Y components, all the X ones can be added together to give the total X force. Same with the Y components.

Since we are using 360 degree notation for the angle, the formula will always give the correct signs. To get components;

Step 2. Get totals in X and Y.

This is the easy bit. Just add the numbers from step 1. You can only add all the Fx's together, and all the Fy's together. The standard symbol for total is sigma (a Greek letter):  In Maths it means "the sum of".

The following maths sentence reads like this: Taking upwards as positive, the sum of all forces in the X direction = Force1 X component + Force2 X component + Force3 X component + ...  

Step 3. Convert back to a single force (resultant).

Now do the reverse. Using Pythagoras theorem, get the length of the hypotenuse with sides Fx and Fy. This is the MAGNITUDE of the resultant: R

To get the ANGLE of the resultant, we take the inverse tangent of Fy/Fx

Seems like hard work, but it is perfect for computers. This is the best method for adding forces on a spreadsheet (Excel) or by any other programming method (Visual Basic, Flash, Javascript etc)

Note 1: Always draw a triangle when determining the resultant. You can't trust you calculator because it can't determine 360 degree notation from the inverse tangent function.

Note 2: Inverse tangent is also called; atan, arctan, invtan, or tan-1. On your calculator it is usually... SHIFT TAN.


Worked Example

Example: A Lifting Eye

Two ropes are attached to this lifting eye. Force A is 1200N at 75o, and Force B is 1600N at 60o from horizontal. 

What is the RESULTANT of these two forces?

Mathematical (components) method
Step 0: Convert angles to 360 Notation:
F1 = 1200 N at 75o
F2 = 1600 N at 120o

Step 1: Get X and Y components:
F1x = 1200 * cos(75) = 310.6 N
F1y = 1200 * sin (75) = 1159 N
F2x = 1600 * cos(120) = -800 N
F2y = 1600 * sin (120) =  1386 N

Step 2: Add X's and Y's
TotX = F1x + F2x = -489.4 N
TotY = F1y + F2y = 2545 N

Step 3:  Convert back to get Mag & Direction;
R = (-489.42 + 25452) ^ 0.5 = 2591 N
Angle = atan (2545 / -489.4) = -79.1o  

  • Take care with the final angle conversion.  This angle is in the 2nd quadrant, so in 360 notation it is; 180 - 79.1 = 100.9o.
  • Obviously this method is silly for 2 forces, but competitive with about 4 forces.

Here is a printout from a force adding program I wrote using VisualBasic. It uses the components method for adding forces.
Triangular Geometry Method
Since there are only 2 forces, we could also do this using triangle geometry. It is not a right-angle triangle, so we need the Cosine rule.

a2 = b2 + c2 - 2*b*c*Cos (A)      Where:
b = 1200
c = 1600
A =  75 + 60 = 135o

Putting these numbers into the formula we get...
a2 = 6715290, so a = 2591.4 N

Now use the sine rule to get angle C
a/sin(A) = c/sin(C) = 3664.77
So sin(C) =  0.436589  
So angle C = 25.886o

So the full angle for resultant is 75 + C = 100.89o

  • This method could be used to add a larger number of forces by doing two forces at a time. There are better alternatives.

Use capital letters for angles and lower case for the length of the opposite sides.

Worked Example 2 (Spreadsheet)

Worked Example 3.9(e) from Adding forces.pdf.

Excel hints

1. Convert to radians: Radians = Degs * pi()/180
2. Now get components...
Fx = Magnitude * cos (Angle)
Fy = Magnitude * sine (Angle)

3. Now add these together to get Rx and Ry


4. Get the Resultant magnitude

= ( F6^2 + G6^2 ) ^0.5                     The square root is "to the power of... 0.5"

5. Get the Resultant angle

= ATAN ( G6 / F6 )

6. Convert back to degrees
Degrees = Radians * 180 / pi()

Adding Forces using AutoCad

Adding Forces using AutoCad (AutoCad 2014)


To enter lines into AutoCad in 360 notation: (For example, 1600 at 120 degrees)

Type 1600 SHIFT < 120 Enter


Whiteboard Photos


Adding forces.pdf. (Ivanoff old edition)
Assignment: Do all questions 3.9(a) to (e) (New edition: Q 4.9) both graphically and mathematically.
Dimension the lengths and angles of all forces CCW from the positive X direction (which is 0 degrees). Show the direction of all forces and also the starting point (small circle). Make the resultant a dotted line (centre-line is typical). A force can be specified like this;  120kN@225o
Solutions 3.9