In this explainer, we will learn how to solve problems about the resolution of a force into two directions.
Force is a vector quantity and so a force can be represented by an arrow in the direction of the force with a length proportional to the magnitude of the force.
The direction of a force can be expressed in terms of a coordinate system. Probably the most intuitive example of such a system is a two-dimensional system with perpendicular axes. It is a common convention to label these axes and , as shown in the following figure.
The direction of a force may be parallel to one of the axes of the coordinate system. This is shown in the following figure for a force for which the point of action is the origin of the system.
When the line of action of a force is parallel to one axis of such a coordinate system, it is necessarily perpendicular to the other axis of the system.
The direction of force can however make an arbitrary angle with lines parallel to either of the axes of the system, as shown in the following figure.
The line of action of the force shown makes an angle with the -axis of the system and makes an angle with the -axis of the system.
A force acting in an arbitrary direction can be expressed in terms of two components. Each component is parallel to one of the axes of the system and perpendicular to the other axis. The directions of these components are, therefore, perpendicular to each other. The perpendicular components of a force and the force itself are shown in the following figure.
The magnitudes of the perpendicular components of a force can be determined from trigonometric rules for right triangles. Consider the following figure of a right triangle which has an angle .
The ratios of the lengths of the opposite and adjacent sides to the length of the hypotenuse are determined by the following equations: and
A force, , and its perpendicular components form a right triangle, as shown in the following figure.
The arrows representing the force and its components are assumed to be in a coordinate system where angle is the angle from the -axis of the system.
The ratios of the magnitudes of the components of the force to the force are, therefore, given by hence, and hence,
The perpendicular components of a force are often represented acting at the point at which the force acts. The following figure shows that representing the components of the force in this way is equivalent to representing the force as the sum of its perpendicular components.
Let us look at an example of resolving a force into perpendicular components.
Example 1: Resolving a Force into Two Perpendicular Components
Resolve a force of 81 N into two perpendicular components and as shown in the figure. Give your answer correct to two decimal places.
Answer
The component is given by
To two decimal places, .
The component is given by
To two decimal places, .
It is worth noting that the magnitudes of the components of a force are dependent on the coordinate system chosen to represent force vectors.
For example, a force that acts along the -axis of a coordinate system has two nonzero components in another coordinate system, as shown in the figure for a coordinate system that is rotated by an angle to the coordinate system in which is parallel to the -axis.
The lines of action of the components and are along the - and -axes of this rotated coordinate system.
Let us look at an example of resolving a force into perpendicular components relative to an arbitrary direction.
Example 2: Resolving a Particle’s Weight into the Directions Parallel and Perpendicular to the Plane
A body weighing 72 N is placed on a plane that is inclined at to the horizontal. Resolve its weight into two components and , where is the component in the direction of the plane and is the component normal to the plane.
Answer
The following figure shows the forces acting on the body, where is the 72 N weight of the body.
The components of the weight parallel to and perpendicular to the plane are perpendicular to each other; hence, and
Let us now look at another such example.
Example 3: Resolving the Weight of a Body on an Inclined Plane
A particle weighing 69 N is placed on a plane inclined at an angle to the horizontal, where . Resolve the weight of the particle into two components, and , where is parallel to a line of greatest slope and is perpendicular to .
Answer
The following figure shows the force of the weight of the particle and its components parallel to and perpendicular to the plane. A section of the plane that corresponds to a right triangle is shown.
The question states that
From this, we can see that the sides of the triangle opposite to and adjacent to the angle have lengths in the ratio . In a right triangle with the sides having this ratio of lengths, the ratio of the side adjacent to to the length of the hypotenuse is , and the ratio of the side opposite to the length of the hypotenuse is .
From this we see that
The component of the weight of the particle perpendicular to the plane, , is rotated through an angle from the direction of the weight of the particle, so has a magnitude given by
Conversely, the component of the weight of the particle parallel to the plane, , has a magnitude given by
Thus far we have considered components of a force where the components are perpendicular to each other. We can consider a force as consisting of two nonperpendicular components. The following figure shows a force and two nonperpendicular components and , where
For a force , there are infinite pairs of components that sum to . and are only one example of such a pair of components.
For nonperpendicular components of a force, it is necessary to use different trigonometric rules to those of right triangles.
The components and can be represented as acting from the same point as , as shown in the following figure.
Lines from the head of to the head of and from the head of to the head of can be drawn. These lines complete a parallelogram, as shown in the following figure.
The line from the head of to the head of is parallel to . The line from the head of to the head of is parallel to . The line of action of is a line from which coangles can be defined for the parallelogram, as shown in the following figure.
The parallelogram consists of two similar triangles, for which the unknown angle, , is given by as shown in the following figure.
Let us consider one of the triangles of the parallelogram, as shown in the following figure.
The lengths of the sides of a triangle are related to the angles of the triangle by the sine rule where , , and are the lengths of the sides opposite to the angles , , and .
For , , and ,
Let us look at an example of resolving a force that has nonperpendicular components.
Example 4: Resolving a Force into Two Components Shown in a Diagram
A force of magnitude 41 N acts due south. It is resolved into two components as shown on the diagram. Find the magnitudes of and . Give your answer to two decimal places.
Answer
A parallelogram can be drawn with side lengths proportional to and . A straight line can be drawn from the vertex at which these sides meet to the opposite vertex of the parallelogram. This line has a length of 41 length units, as shown in the following figure.
If we consider one of the triangles of this parallelogram, it contains an angle , as shown in the following figure.
The angle is given by
The magnitudes of and can be determined by applying the sine rule in the triangle:
We have, therefore, that
To two decimal places, .
We also have that
To two decimal places, .
Let us look at another such example.
Example 5: Resolving Forces in a Real-Life Context
The diagram shows a body of weight 69 N suspended by 2 light, inextensible strings, and . Both strings make an angle of with the horizontal. Resolve the weight of the body into two components in the direction of and in the direction of . Give your answers to the nearest newton.
Answer
The strings are at the same angle from the horizontal and so the components and have equal magnitude. The line of action of the weight of the body is perpendicular to .
The following figure shows two right triangles. For each triangle, the non-right angles are and .
The angle between each component and the weight is also and so is given by
A parallelogram can be defined that has a vertex at and another vertex vertically below proportional to the weight, where the lengths of each side of the parallelogram are proportional to the magnitude of either component. A triangle of this parallelogram is shown in the following figure.
From this, we can see that
To the nearest newton, .
Each component has a magnitude of 57 N.
Now let us summarize what we have learned in these examples.
Key Points
- A two-dimensional force, , can be resolved into components, and . If these components are perpendicular to each other in a coordinate system used to represent the force, then these components have magnitudes given by and where is the angle between and the -axis of the coordinate.
- A force can be resolved into components that are not perpendicular to each other. A parallelogram can be formed that has a diagonal of length proportional to the magnitude of the force and sides that have lengths proportional to the magnitudes of the components of the force that act at angles to the force proportional to the internal angles of the parallelogram. The lengths of the sides of the parallelogram can be determined using the sine rule where is a triangle of the parallelogram with side lengths , , and , where , , and are lengths of the sides of the triangle opposite the angles , , and .
