4.7 – Vector Addition | Physics with Mr Shone

A vector quantity is a quantity that is fully described by both its magnitude and direction whereas a scalar quantity is a quantity that is fully described by its magnitude.
Forces, velocity and acceleration are examples of vector quantities.

Unbalanced Forces

If forces are not balanced it will result in a resultant force (net force) being present on the object.

The resultant force acting on an object can be determined using the principles of vector addition.

Addition of Parallel Forces

Parallel forces the force vectors are simply added together.
Antiparallel forces (i.e. directions 180º apart) can be dealt with by subtraction.

Example 1: Parallel Forces (Horizontal Forces)
What is the resultant force acting on the box? As the forces are all parallel to each other we can simply add/subtract to find the resultant. In this case: F_resultant = 45 N + 20 N – 15 N = 50 N to the right

Example 1: Parallel Forces (Horizontal Forces)

What is the resultant force acting on the box?

As the forces are all parallel to each other we can simply add/subtract to find the resultant.

In this case:

F_resultant = 45 N + 20 N – 15 N = 50 N to the right

Example 2: Parallel Forces (Vertical Forces)
What is weight of the steel rod? Questions like this we should first identify all of the forces acting on the object (and preferably draw a free-body diagram too). There are only three forces acting on the rod the tension at each end (both acting upwards) and weight of the steel rod acting downwards. Thus weight = 2.0 N + 2.5 N = 4.5 N downwards

Example 2: Parallel Forces (Vertical Forces)

What is weight of the steel rod?

Questions like this we should first identify all of the forces acting on the object (and preferably draw a free-body diagram too).

There are only three forces acting on the rod the tension at each end (both acting upwards) and weight of the steel rod acting downwards.

Thus weight = 2.0 N + 2.5 N = 4.5 N downwards

Example 3: Parallel Forces (Maximum and Minimum Resultant Force)
Determine the maximum and minimum possible resultant force that can be obtained from the addition of 3.0 N and 4.0 N. Maximum resultant force (R_MAX) will occur when the forces are aligned in the same direction. Minimum resultant force (R_MIN) will occur when the forces are aligned in opposite directions.

Addition of Non-Parallel Forces

Non-parallel forces can be added by drawing a scaled diagram using an appropriate scale based on the two following drawing methods:

- Parallelogram Method
- Vector Triangle Method

Alternatively solutions can also be calculated using mathematical (trigonometric) methods if preferred.

Method 1: Parallelogram Method

Applying the Parallelogram Method to solve a problem:

(a) Draw 2 vectors starting from the same point O
(b) Complete the parallelogram
(c) Draw the resultant vector from O along the diagonal.

Advantages of this method: Simple & Quick

Disadvantages of this method: can only be used to add 2 vectors at once.

Addition of Forces: Parallelogram Method
What is the magnitude and direction of the resultant of the following two forces? Draw a scale diagram with the two forces 35º apart. Use these two lines to construct a parallelogram. The diagonal between them represents the resultant. With a scale of 1 cm represents 1 N, the resultant force is found to be 17.2 N. The direction of the force is 13º anticlockwise from the 11.0 N force.

Addition of Forces: Parallelogram Method

What is the magnitude and direction of the resultant of the following two forces?

Draw a scale diagram with the two forces 35º apart.
Use these two lines to construct a parallelogram.
The diagonal between them represents the resultant.

With a scale of 1 cm represents 1 N, the resultant force is found to be 17.2 N.

The direction of the force is 13º anticlockwise from the 11.0 N force.

Method 2: VectorTriangle (Polygon) Method

Applying the VectorTriangle (Polygon) Method to solve a problem:

(a) Draw one vector from A, followed by the second vector ending at B.
(b) Draw the resultant vector starting at A and ending at B.

Advantages of this method: Can be used to add any number of vectors at once.

Note
Scale diagram should be labelled for: Magnitude or symbol/name of vectors Direction of vectors using arrows (double arrows for resultant) Indicate the arrow in the middle of the line (not at the end!) Any known angle(s)

Note

Scale diagram should be labelled for:

- Magnitude or symbol/name of vectors
- Direction of vectors using arrows (double arrows for resultant)
- Indicate the arrow in the middle of the line (not at the end!)
- Any known angle(s)

Addition of Forces: Vector Triangle Method
An object O is under the action of two forces. 500 N acts north and 350 N in a direction 60° west of north (or 30° north of west).Determine the magnitude and direction of the resultant force by drawing a scale diagram using the vector triangle method. Step 1: Decide on a suitable scale that will give you a large diagram in the space provided and write down this scale clearly. Scale used: 1.0 cm represents 50 N Step 2: Draw one of the forces making sure to keep the orientation the same as in the question. Following the chosen scale, this line will be drawn 10.0 cm long. Step 3: Draw the second force arrow starting off from the tip of the first force arrow. Ensure that the angle of the force remains unchanged. (Here a protractor is used to ensure the arrow is drawn at 60° from the vertical). Following the chosen scale, this line will be drawn 7.0 cm long. If there are more than 2 forces, connect them all in this same manner (i.e. head-to-tail) – You will not end up with a triangle, but will always end up with a closed polygon. Step 4: Complete the diagram by drawing the resultant from the start of the first arrow to the tip of of the last arrow. Measure the length of the arrow to get the magnitude of the force. In this example the red line is 14.8 cm in length => R=740 N Measure an angle to relate the angle of this resultant force to one of the other forces. In this example: the resultant force acts an angle 24° west of north.

Addition of Forces: Vector Triangle Method

An object O is under the action of two forces. 500 N acts north and 350 N in a direction 60° west of north (or 30° north of west).

Determine the magnitude and direction of the resultant force by drawing a scale diagram using the vector triangle method.

Step 1: Decide on a suitable scale that will give you a large diagram in the space provided and write down this scale clearly.

Scale used: 1.0 cm represents 50 N

Step 2: Draw one of the forces making sure to keep the orientation the same as in the question.
Following the chosen scale, this line will be drawn 10.0 cm long.

Step 3: Draw the second force arrow starting off from the tip of the first force arrow. Ensure that the angle of the force remains unchanged. (Here a protractor is used to ensure the arrow is drawn at 60° from the vertical).
Following the chosen scale, this line will be drawn 7.0 cm long.

If there are more than 2 forces, connect them all in this same manner (i.e. head-to-tail) – You will not end up with a triangle, but will always end up with a closed polygon.

Step 4: Complete the diagram by drawing the resultant from the start of the first arrow to the tip of of the last arrow.

Measure the length of the arrow to get the magnitude of the force.

In this example the red line is 14.8 cm in length => R=740 N

Measure an angle to relate the angle of this resultant force to one of the other forces.

In this example: the resultant force acts an angle 24° west of north.

Balanced Forces

When forces acting on an object are balanced, the resultant force on the object is zero and it will not accelerate.
Such objects are said to be in equilibrium.
Three non-parallel forces acting on an object at equilibrium will form a closed triangle with the arrows pointing in the clockwise or anticlockwise direction.

To sketch a vector diagram:
(a) Draw any force first from a point A (b) Then draw the 2^nd force followed by the 3^rd force ending at A again.

Note
The order in which we draw the forces is not important.

Example: Vector Diagram for an object in Equilibrium
Three forces F₁, F₂ and F₃ act on a point to keep it in equilibrium as shown below. Draw a vector diagram to show the addition of these 3 forces. Knowing that the object is in equilibrium we know that the resultant force is zero and so a force addition diagram will result in a closed triangle. Starting from a point A we can first draw F₁ followed by F₂ and then F₃ and we will end up with the following closed triangle. Alternatively we could have started from A and drawn F₁ followed by F₃ and then F₂ and we would still end up with a closed triangle. You can probably figure out what the triangles would look like for other orders of drawing the 3 forces.

Example: Vector Diagram for an object in Equilibrium

Three forces F₁, F₂ and F₃ act on a point to keep it in equilibrium as shown below.

Draw a vector diagram to show the addition of these 3 forces.

Knowing that the object is in equilibrium we know that the resultant force is zero and so a force addition diagram will result in a closed triangle.

Starting from a point A we can first draw F₁ followed by F₂ and then F₃ and we will end up with the following closed triangle.

Alternatively we could have started from A and drawn F₁ followed by F₃ and then F₂ and we would still end up with a closed triangle.

You can probably figure out what the triangles would look like for other orders of drawing the 3 forces.

To find an unknown force:
(a) Sketch the vector diagram. (b) Calculate using trigonometry formulae or draw scale diagram. (c) Describe the direction of the force.

Comparison of vector triangles

These two triangles may look similar, but are in fact showing different concepts:

Unbalanced Forces

Balanced Forces

Adding 2 forces acting on a body to give a resultant force R on the body.

Adding 3 forces acting on a body in equilibrium to give zero resultant force.

Angled Forces
Compare the forces T₁ and T₂. a) Will T₁ be equal in magnitude to T₂? No. The forces will be of differing magnitudes. b) Will T₁ and T₂add up to 100 N? No. The sum will always be larger than 100 N. c) Will the vertical components of the T₁ and T₂ be equal? No. The vertical components of T₁ and T₂ will add up to 100 N. d) Will the horizontal components of the T₁ and T₂ be equal? Yes. The horizontal components of T₁ and T₂ will be of equal magnitude but act in opposite directions. The net horizontal force must be zero.

Angled Forces

Compare the forces T₁ and T₂.

a) Will T₁ be equal in magnitude to T₂?

No. The forces will be of differing magnitudes.

b) Will T₁ and T₂add up to 100 N?

No. The sum will always be larger than 100 N.

c) Will the vertical components of the T₁ and T₂ be equal?

No. The vertical components of T₁ and T₂ will add up to 100 N.

d) Will the horizontal components of the T₁ and T₂ be equal?

Yes. The horizontal components of T₁ and T₂ will be of equal magnitude but act in opposite directions. The net horizontal force must be zero.