## How to find resultant force of three forces

Hence a single force which can replace a number of forces acting on a rigid body, without causing any change in the external effects on the body, is known as the resultant force. Collinear Coplanar forces are those forces which act in the same plane and have a common line of action. The resultant of these forces are obtained by analytical method or graphical method. The resultant is obtained by adding all the forces if they are acting in the same direction. If any one of the forces is acting in the opposite direction, then resultant is obtained by subtracting that force.

Their resultant R will be sum of these forces. Some suitable scale is chosen and vectors are drawn to the chosen scale. In Fig. Then the length ad represents the magnitude of the resultant on the scale chosen. This resultant is shown in Fig. This force is acting from a to b. The force 1 F 2 is taken equal to be on the same scale in opposite direction.

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This force is acting from b to c. The force Fa is taken equal to cd. This force is acting from c to d. The resultant force is represented in magnitude by ad on the chosen scale. Concurrent coplanar forces are those forces which act in the same plane and they intersect or meet at a common point.

We will consider the following two cases:. The magnitude of resultant is obtained from equation 1. Suppose two forces P and Q act at point 0 as shown in Fig. Let e is the angle made by the resultant R with the direction of force P. Forces P and Q form two sides of a parallelogram and according to the law, the diagonal through the point O gives the resultant R as shown.

The resultant can also be determine graphically by drawing a triangle oac as explained below and shown in Fig. The resultant of three or more forces acting at a point is found analytically by a method which is known as rectangular components methods. Then resultant R is given by. The force F 1 is resolved into horizontal and vertical components and these components are shown in Fig. Similarly, Figs. The various horizontal components are:. The resultant of several forces acting at a point is found graphically with the help of the polygon law of forces, which may be stated as.

The resultant 1s obtained graphically by drawing polygon of forces as explained below and shown in Fig. From a, draw vector ab parallel to force OF1.In mechanics we deal with two types of quantities variables : scalar and vector variables. Scalar variables have only magnitude, for example: length, mass, temperature, time. Vector variables have magnitude and direction, for example: speed, force, torque.

The direction of the vector is defined by the angles of the force witch each axis. The vector variables are usually represented using bold symbols with arrows on top. Several forces can act on a body or point, each force having different direction and magnitude.

In engineering the focus is on the resultant force acting on the body. The resultant of concurrent forces acting in the same plane can be found using the parallelogram lawthe triangle rule or the polygon rule.

Two or more forces are concurrent is their direction crosses through a common point. For example, two concurrent forces F 1 and F 2 are acting on the same point P. In order to find their resultant Rwe can apply either the parallelogram law or triangle rule. If there are several forces acting on the same point, we can apply the polygon rule to find their resultant. The resultant force can be determined also for three-dimensional force systemsby using the polygon rule. The parallelogram law, triangle rule and polygon rule are geometric methods to find the force resultant.

The law of sines gives the relationship between the forces and the angles :. The resultant force can also be calculated analyticalusing force projections. Using the force projection methodwe can calculate the magnitude and direction angles of the resultant force. Replacing 4 in the equations above gives the angles with each axis as trigonometric functions :.

The force projection method can also be used for co-planar x, y-axis force resultant calculations.

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Example 1. Step 1. To get an idea on how the resultant force might look like, we can apply to polygon rule. As you can see, the magnitude of the resultant is nearly equal with that of the force F 3. This geometrical solution is helpful because we know what results we should expect from the analytical solution. As expected, the analytic solution forces projection give the same results as the geometric solution polygon rule.

Example 2. For two-dimensional problems, we can write down the general equations to calculate the vertical and horizontal force components as:. Example 3.

Observation : If the calculation is done on hand-held calculator of a software application, the argument of the cos function must be given in radians, for example:. Observation : If the calculation is done on hand-held calculator of a software application, the argument of the sin function must be given in radians, for example:. This method can be extended to any number of forces, as long as the force values and angles are known. Use the calculator above to calculate and evaluate different distribution of forces.

By hovering the mouse pointer on the line forces, you can see their coordinates which represent the F x 5812 and F y 5812 components. Hello, I would like to use two parallelogram-of-forces figures from your website for a publication. Please can you approve the use of the figures? Hi Susanne, You can use the figure as long as the source is stated.In this article, you will learn what the resultant force also known as net force is, and how to find it when an object is subject to parallel forces as well as non-parallel forces with the help of examples.

When an object is subject to several forces, the resultant force is the force that alone produces the same acceleration as all those forces. The reason why the resultant force is useful is that it allows us to think about several forces as though they were a single force.

This means that to determine the effect that several forces have on an object, we only need to determine the effect that a single force has. If we know the mass m of an object and the acceleration a produced by the forces that act on it, we can find the resultant force using Newton's Second Law. Indeed, according to Newton's Second Law, the force F that alone produces the acceleration a on an object of mass m is:. Which indicates that the resultant force R has the same direction as aand has magnitude equal to the product m a.

For example, if a box of 1. Often, however, we know the forces that act on an object and we need to find the resultant force. Experiments show that when an object is subject to several forces, F 1F 2Notice that this is not a mere sum of the magnitudes of the forces, but the sum of the forces taken as vectorswhich is more involved because vectors have both a magnitude and a direction that we need to consider when doing the sum.

According to the above equation, if an object is subject to no forces, then the resultant force is zeroand if an object is subject to only one force, then the resultant force is equal to that force.

These two cases are pretty simple, but what about an object subject to two or more forces? How do we perform the vector sum then? To explain this clearly, we will now go through all the cases that can happen, from simple ones in which all the forces are parallel, to more complex ones in which the forces are not parallel, and show how to find the resultant force in each of them with the help of examples.

Let's start with the simple case in which an object is subject to two forces that act in the same direction:. The resultant force is in the same direction as the two forcesand has the magnitude equal to the sum of the two magnitudes :. Let's consider the case in which an object is subject to two forces that act in opposite directions. The resultant force will be zero because two opposite forces cancel each other out. To find the resultant force in this case, we first sum all the forces that go in one direction, and then all the forces that go in the other direction:.

In the previous cases, we have forces that are all parallel to one another. It's time to consider the case in which an object is subject to two forces that are not parallel. For example, let's assume that we have a block subject to two forces, F 1 and F 2. Since one of the two forces is horizontal, for convenience, we choose the x -axis horizontal, and the y -axis vertical, and we place the origin at the center of our block:.

The next step is to determine the x and y components of all the forces that act on the block :. If we sum all the x components, we will get the x component of the resultant force :. Similarly, if we sum all the y components, we will get the y component of the resultant force :. At this point, we know the x and y components of Rwhich we can use to find the magnitude and direction of R :.

The magnitude of R can be calculated by applying Pythagoras' Theorem :. Finally, let's examine the case in which an object is subject to more than two non-parallel forces. For example, suppose we have an object that is subject to three forces, F 1F 2and F 3.

We can find the resultant force R using the same process that we used in the previous case of two non-parallel forces.In Unit 2 we studied the use of Newton's second law and free-body diagrams to determine the net force and acceleration of objects.

In that unit, the forces acting upon objects were always directed in one dimension. There may have been both horizontal and vertical forces acting upon objects; yet there were never individual forces that were directed both horizontally and vertically. Furthermore, when a free-body diagram analysis was performed, the net force was either horizontal or vertical; the net force and corresponding acceleration was never both horizontal and vertical.

Now times have changed and you are ready for situations involving forces in two dimensions. In this unit, we will examine the effect of forces acting at angles to the horizontal, such that the force has an influence in two dimensions - horizontally and vertically.

For such situations, Newton's second law applies as it always did for situations involving one-dimensional net forces. However, to use Newton's laws, common vector operations such as vector addition and vector resolution will have to be applied.

In this part of Lesson 3, the rules for adding vectors will be reviewed and applied to the addition of force vectors. Methods of adding vectors were discussed earlier in Lesson 1 of this unit. During that discussion, the head to tail method of vector addition was introduced as a useful method of adding vectors that are not at right angles to each other. Now we will see how that method applies to situations involving the addition of force vectors.

A force board or force table is a common physics lab apparatus that has three or more chains or cables attached to a center ring. The chains or cables exert forces upon the center ring in three different directions. Typically the experimenter adjusts the direction of the three forces, makes measurements of the amount of force in each direction, and determines the vector sum of three forces. Forces perpendicular to the plane of the force board are typically ignored in the analysis. Suppose that a force board or a force table is used such that there are three forces acting upon an object.

The object is the ring in the center of the force board or force table. In this situation, two of the forces are acting in two-dimensions. A top view of these three forces could be represented by the following diagram. The goal of a force analysis is to determine the net force and the corresponding acceleration.

The net force is the vector sum of all the forces. That is, the net force is the resultant of all the forces; it is the result of adding all the forces together as vectors. One method of determining the vector sum of these three forces i. In this method, an accurately drawn scaled diagram is used and each individual vector is drawn to scale. Where the head of one vector ends, the tail of the next vector begins. Once all vectors are added, the resultant i. This procedure is shown below.Resultant force — Vector diagrams of forces: graphical solution.

We call a force that can replace two or more other forces and produce the same effect a resultant force.

For example, the forces F 1 and F 2 acting in the same direction on the point P along the same line of action can be replaced by a single force having the same effect as shown in Fig. Since both the forces F 1 and F 2 have a common line of action and act in the same direction, their vector lengths can simply be added together to obtain the magnitude of the vector of the single force that can replace them. This is the force F R shown in the Fig b. It is called the resultant force.

Note that the vector for a resultant force has a double arrowhead to distinguish it from the other forces acting in the system. F R could have been determined by simply subtracting the vector length for F 1 from the vector length for F 2 as shown in c. When two or more forces whose lines of action lie at an angle to each other act on a single point P, their resultant force cannot be determined by simple arithmetical addition or subtraction.

If only two forces are involved, as shown in below figure, the magnitude and direction of the resultant force can be determined by drawing, to scale, a parallelogram of forces. Figure b shows how these vectors form two adjacent sides of a parallelogram PABC.

A force which cancels out the effect of another force or system of forces is called the equilibrant force F E. An equilibrant force:. Figure iii below shows the equilibrant force F E to be equal and opposite to the resultant force F Rdetermined in Fig.

The resolution of forces is the reverse operation to finding the resultant force. That is, the resolution of a force is the replacement of a given single force by two or more forces acting at the same point in specified directions. To resolve the force into its horizontal and vertical component forces first draw the lines of action of these forces from the point P as shown in Fig. Then complete the parallelogram of forces as shown in Fig.

The magnitude of the forces F V and F H can be obtained by scaling the drawing vector diagram or by the use of trigonometry. Figure v shows three coplanar forces acting on a body. Their lines of action are extended backwards to intersect at the point P. The point P is called the point of concurrency.

Figure vi shows a further worked example. The accuracy of the answers depends on the accuracy of the drawing. This should be as large as possible to aid measurement.

The polygon of forces is used to solve any number of concurrent, coplanar forces acting on a point P. They are concurrent because they act on a single point P. Figure vii — a shows a space diagram with the spaces between the forces designated by capital letters. The corresponding force diagram is shown in Fig. As their name implies, non-concurrent coplanar forces lie in the same plane but do not act at a single point point of concurrency.

An example is shown in Fig. Note, although the resolution of forces and systems of forces by graphical methods is simple and straightforward, the accuracy depends on the scale of the drawing and the quality of the draughtsmanship.

Where more accurate results are required they can be obtained by mathematical calculation using trigonometry. Resultant force Resultant force — Vector diagrams of forces: graphical solution resultant force — It has already been stated that forces cannot be seen so they cannot be drawn; however, the effect of forces and systems of forces can be represented by vectors. Resultant forces We call a force that can replace two or more other forces and produce the same effect a resultant force.

Parallelogram of forces When two or more forces whose lines of action lie at an angle to each other act on a single point P, their resultant force cannot be determined by simple arithmetical addition or subtraction. Equilibrant forces A force which cancels out the effect of another force or system of forces is called the equilibrant force F E. An equilibrant force: has the same magnitude as the resultant force, has the same line of action as the resultant force, acts in the opposite direction to the resultant force.

Forces that act in the same plane are said to be coplanar.All the guesthouses we stayed in were very very comfortable. Much better than we had expected and the owners were all lovely.

## Resultant forces – three or more forces at an angle

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Adding Vectors: How to Find the Resultant of Three or More Vectors

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### How to calculate the resultant force acting on an object

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