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# Resolving Forces

You can add forces together to give a resultant force. Did you know that a single force can also be broken down into component forces at right angles to each other?

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Resolving forces is the process of finding two or more forces that, when combined, will produce a force with the same magnitude and direction as the original.

Vectors can have two parts when directed at an angle to the customary coordinate axis. Each of these is directed along with one of the axes, either horizontally or vertically. The process of breaking down a force into its Cartesian coordinate components is a common task when solving statics problems.

If a force pulled a particle upwards and to the right, that single force could be resolved into two separate components. The one directed upwards (vertical components), and the other directed right (horizontal component). This process can be done with the help of trigonometric functions.

Let's assume that the force exerted on the particle is 60N and is at an angle of 40 degrees above the horizontal. We can model this in the figure below to help resolve our force into two significant components.

Resolving forces example

With this example, we will have to make projections on our figure to complete the right-angled triangle. From this, we see which sides of the triangles we will find – the sides of the triangles equal to either the horizontal or the vertical components of the force.

Resolving forces projection

Let the horizontal component be

And the vertical component berole="math" style="max-width: none;">

Solving for the horizontal component:

$$\cos 40^\circ = \frac{F_1}{60N}$$

$$F_1 = 60 N \space \cos 40^\circ = 45.9 N$$

Solving for the vertical component:

$$\sin 40^\circ = \frac{F_2}{60 N}$$

$$F_2 = 60 N \space \sin 40^\circ = 38.6N$$

## Resolving concurrent forces in equilibrium

When forces are applied to a body so that their lines of action meet at a point, they are considered concurrent forces. The result of these forces on a body can also be found with the help of trigonometric functions.

Given that the particle below is in equilibrium find the value of A and B.

Resolving forces acting on a particle in equilibrium

First, let's complete the two right-angled triangles opposite the 45º angles (shown below).

Resolving forces acting on a particle in equilibrium

According to trigonometry, the triangle with the side 2AN is a hypotenuse, 2Asin45° N is the side opposite the angle, and 2Acos45° N is the side adjacent to the angle. In the second triangle, AN is the hypotenuse, Asin45° N is the side opposite the angle, and Acos45° N is the side adjacent to the angle.

All forces will be resolved into their horizontal and vertical components separately. Let's start by resolving all of the forces in the diagram vertically. All values of forces that are working upwards are treated as positive values, and those that are working downwards are treated as negative values, as they are vectors.

The sum of upward and downward forces in equilibrium is zero.

2Asin45° N - Asin45° N - 50N = 0

Asin 45° N-50N = 0

Asin 45° N = 50N

$$A = \frac{50 N}{\sin 45^\circ} = 50 \sqrt{2} N$$

Now, we're going to resolve horizontally to find the value of B.

All values of forces that work to the right are considered positive, and those that work to the left are considered negative.

The sum of all forces to the left and right is equal to zero in equilibrium.

2Acos45° N + Acos45° N - B = 0

3Acos45° N = B

We will now substitute A into the equation.

$$3 \cdot 50 \sqrt{2} \cos 45 ^\circ N = B = 150 N$$

## Resolving forces in a truss

A truss is a plane that takes advantage of the inherent geometric stability of triangles to distribute weight in harmony and to handle changing compressions and tension. They are a support system for structures that consist of a web of triangles to distribute pressure and tension evenly. A roof is a very good example of a truss.

There are a couple of steps to finding forces in a truss. Let's look at an example:

To analyse the following truss, you would have to break it down.

Truss

Step 1 . Create a free-body diagram of the entire truss which should include all forces. Ignore the individual triangles and label all distances and known triangles.

The free-body diagram of a truss

Step 2 . We will pick the pivot with the most unknowns and sum all the moments around it. We will pick point A in this case, and the formula here will be $$\sum M = 0$$. The three moments around pivot A are:

• Reaction force at B causing counterclockwise moment.
• 500 lb applied force, causing a clockwise moment.
• 150 lb applied force causing a clockwise moment

Moment = Force x Perpendicular distance

$$(RB_y \cdot 4 ft) - (500 lbs \cdot 2 ft) - (150 lbs \cdot 2 ft) = 0$$

$$RB_y = 325 lbs$$

Moments around pivot A in a truss

Step 3 . Sum all forces in the x-direction and equate it to 0.

$$\sum F_x = 0$$

$$RA_x + 150 lbs = 0$$

$$RA_x = -150 lbs$$

Resolving truss to find x component

Step 4 . Sum all forces in the y-direction and equate it to 0.

$$\sum F_y = 0$$

$$RA_y + RB_y -500 lbs = 0$$

We already found $$RB_y = 325 lbs$$, so we will substitute that into the equation.

$$RA_y = 175 lbs$$

Resolving truss to find y component

Step 5 . We will use the method of joints to solve for tension and compression for each member since we now know what the three reaction forces are. Now, create a free-body diagram for each joint and label each member of the two endpoints:

Free-body diagram of a joint in a truss

Step 6. We will now use trigonometric functions to resolve diagonal vectors into x and y components.

$$BD_y = (0.894)BD$$

$$BD_x = (0.448) BD$$

resolving forces in a joint in a truss

Step 7 . Sum all forces in the y-direction and equate it to 0.

$$\sum F_y = 0$$

$$(0.894)BD + 325 lbs = 0$$

$$BD = -363.5 lbs$$

Step 8. Sum all the forces in the x-direction and equate it to 0.

$$\sum F_x = 0$$

$$BC - 0.448 \cdot BD = 0$$

$$BC = 162.9 lbs$$

Step 9. You can now repeat steps 5 through to 8 for every joint.

## Resolving Forces - Key takeaways

• A single force can be broken down into component forces at right angles to each other.
• Resolving forces is the process of finding two or more forces that, when combined, will produce a force with the same magnitude and direction as the original.
• Trigonometric functions help resolve forces into component forms.
• A truss is a plane that takes advantage of the inherent geometric stability of triangles. It distributes weight in harmony and handles changing compressions and tension.
• To resolve a force, make projections on your diagram to form right-angled triangles and use trigonometric functions to find the unknown x and y components.

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##### Frequently Asked Questions about Resolving Forces

What are resolving forces?

Resolving forces is the process of finding two or more forces that, when combined, will produce a force with the same magnitude and direction as the original.

How do you calculate resolving forces?

First make projections on your diagram to form right-angled triangles and use trigonometric functions to find the unknown x and y components.

How do you resolve forces on an inclined plane?

Project your angle of incline from the origin of the unbalanced force acting on the particle. Now you can have your right-angled triangle in addition to a known angle to help resolve forces using trigonometry.

Into how many possible components can a single force be resolved?

Two components

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