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Resolving Forces

- Calculus
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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?

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.

Answer:

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.

Let the horizontal component be

And the vertical component be

Solving for the horizontal component:

Solving for the vertical component:

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.

Answer:

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

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

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.

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.

Answer:

**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.

**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 . 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

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

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

We already found , so we will substitute that into the equation.

**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:

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

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

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

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

- 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.

Two components

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