Forces as Vectors

Definition of Vectors

A vector is a quantity that has both magnitude (size) and direction. Forces are vectors because they push or pull in a specific direction and have a strength measured in newtons (N).

For example, a force of 10 N acting to the right is different from a force of 10 N acting upwards, even though the magnitude is the same.

Magnitude and Direction

The magnitude of a force is how strong it is, measured in newtons (N). The direction is the line along which the force acts, indicated by an arrow.

Both magnitude and direction are essential to fully describe a force. Changing either changes the effect of the force.

Representation by Arrows

Forces are often represented by arrows in diagrams:

• The length of the arrow shows the magnitude of the force (longer arrow = bigger force).
• The direction of the arrow shows the direction in which the force acts.

This visual representation helps to understand how forces combine and interact.

Example:

A force of 5 N acting to the right can be drawn as an arrow pointing right with a length proportional to 5 N. A force of 10 N acting upwards would be an arrow twice as long pointing upwards.

Vector Addition of Forces

When two or more forces act on an object, they combine to form a single force called the resultant force. This resultant force determines the overall effect on the object.

Adding Forces in the Same Direction

If forces act along the same line and in the same direction, add their magnitudes to find the resultant force.

For example, if two forces of 3 N and 7 N act to the right, the resultant force is:

$3\,\text{N} + 7\,\text{N} = 10\,\text{N}$ to the right.

Adding Forces in Opposite Directions

If forces act along the same line but in opposite directions, subtract the smaller force from the larger one. The resultant force acts in the direction of the larger force.

For example, if 8 N acts to the right and 5 N acts to the left, the resultant force is:

$8\,\text{N} - 5\,\text{N} = 3\,\text{N}$ to the right.

Example:

Two people pull a box with forces of 12 N and 15 N in the same direction. The resultant force is:

$12\,\text{N} + 15\,\text{N} = 27\,\text{N}$ in that direction.

If the 15 N force acts to the right and the 12 N force acts to the left, the resultant force is:

$15\,\text{N} - 12\,\text{N} = 3\,\text{N}$ to the right.

Resolving Forces

Forces can act at angles, not just along the same line. To analyse these, we resolve a force into two perpendicular components, usually horizontal and vertical.

This means breaking one force into two forces at right angles that have the same combined effect as the original force.

Components of Forces

If a force $F$ acts at an angle $\theta$ to the horizontal, its components are:

• Horizontal component: $F_x = F \cos \theta$
• Vertical component: $F_y = F \sin \theta$

These components are easier to work with when calculating net forces or checking if an object is in equilibrium.

Using Perpendicular Vectors

Because the components are at right angles, they can be treated independently. The original force can be found again by combining the components using Pythagoras’ theorem:

$$F = \sqrt{F_x^2 + F_y^2}$$

The angle $\theta$ can be found using:

$$\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)$$

Applications in Equilibrium

When an object is in equilibrium (not moving or moving at constant velocity), the sum of forces in both horizontal and vertical directions is zero.

Resolving forces helps to check this condition by ensuring:

• $\sum F_x = 0$
• $\sum F_y = 0$

Example:

A force of 20 N acts at 30° above the horizontal. Its horizontal and vertical components are:

Horizontal: $20 \times \cos 30^\circ = 20 \times 0.87 = 17.4\,\text{N}$

Vertical: $20 \times \sin 30^\circ = 20 \times 0.5 = 10\,\text{N}$

Vector Diagrams

Vector diagrams are used to represent forces graphically to scale. This helps to find the resultant force when forces act at angles.

Drawing Force Vectors to Scale

Choose a scale, for example 1 cm = 5 N. Draw each force as an arrow with length proportional to its magnitude and in the correct direction.

Using Scale Diagrams

To find the resultant force:

1. Draw the first force vector to scale in the correct direction.
2. From the tip of this vector, draw the second force vector to scale in its direction.
3. Draw a straight line from the start of the first vector to the end of the second vector.
4. This line represents the resultant force vector.
5. Measure its length and convert back to newtons using the scale.
6. Use a protractor to measure the angle of the resultant force.

Determining Resultant Force Graphically

This method works well for two forces acting at any angle. For more forces or more accuracy, calculations using components are preferred.