Resulting force how it is calculated and exercises solved

The force resulting it is the sum of all the forces that act on the same body. When a body or object is subjected to the action of several forces simultaneously, an effect occurs. Acting forces can be replaced by a single force that produces the same effect. This single force is the resultant force also known as the net force and is represented by the symbol F_R .

The effect it produces F_R it will depend on its size, direction and sense. Physical quantities that have direction and sense are vector quantities.

Being the forces acting on a body vector magnitudes, the resultant force F_R  is a vector sum of all forces and can be represented graphically with an arrow that indicates its direction and direction.

With the resultant force, the problem of a body affected by several forces is simplified by reducing it to a single force acting.

Article index

• 1 Formula
• 2 How do you calculate the resultant force?
  • 2.1 resultant of parallel forces 
  • 2.2 Non-parallel forces
• 3 Solved exercises
• 4 References

Formula

The mathematical representation of the resultant force is a vector summation of the forces.

 F_R= ∑F          (1)

 ∑F = F₁+ F_two+ F₃+... F_N           (two)

F_R= Resulting force

∑F = Sum of Forces

N= Number of Forces

The resultant force can also be represented by Newton's second law equation.

   F_R= m.to          (3)

m= body mass

a = body acceleration

If equation (1) is substituted in equation (3), the following equations are obtained:

∑F = m.to          (4)

F₁+ F_two+ F₃+... F_{N =} m.to          (5)

The mathematical expressions (4) and (5) provide information about the state of the body by obtaining the acceleration vector to.

How do you calculate the resultant force?

The resultant force is obtained by applying Newton's Second Law which states the following:

The net force acting on a body is equal to the product of its mass and the acceleration it acquires. (Equation (3))

The acceleration of the body will have the direction of the applied net force. If all the forces acting on the body are known, it would be enough to add them vectorially to obtain the resultant force. Likewise, if the resultant force is known then it would be enough to divide it by the mass of the body to obtain its acceleration.

If the resultant force is zero, the body is at rest or at constant speed. If a single force acts on the body, the resulting force is equal to that force F_R=F.

When several forces act on the same body, the vector components of the force must be taken into account, and whether these forces are parallel or not.

For example, if we horizontally slide a book placed on a table, the forces in the horizontal direction are the only ones that provide acceleration to the body. The net vertical force on the book is zero.

If the force applied to the book has an inclination with respect to the horizontal plane of the table, the force is written as a function of the vertical and horizontal components.

Resulting parallel forces 

The parallel forces that act on a body are those forces that act in the same direction. They can be of two types of equal or opposite sense.

When the forces acting on a body have the same direction and the same direction or are in the opposite direction, the resulting force is obtained by performing the algebraic sum of the numerical values ​​of the forces.

Non-parallel forces

When non-parallel forces are applied to a body, the resultant of the forces will have rectangular and vertical components. The mathematical expression to calculate the net force is:

F_R^two= (∑ F_x)^two+(∑ F_Y)^two            (6)

so θ_x= ∑ F_Y / ∑ F_x         (7)

∑ F_x  and ∑ F_x= Algebraic summation of the components x and Y of applied forces

θ_x= angle formed by the resultant force F_R with shaft x

Note that the resultant force of expression (6) is not highlighted in bold type and it is because it only expresses the numerical value. The direction is determined by the angle θ_x.

Expression (6) is valid for forces acting in the same plane. When forces act in space, the component is taken into account z of force when working with rectangular components.

Solved exercises

The parallel forces in the same direction are added and subtracted with the parallel force in the opposite direction

F_R= 63 N + 50 N - 35 N = 78N

The resultant force has a magnitude of 78N with horizontal direction.

2. Calculate the resultant force of a body under the influence of two forces F_{1 Y} F_two. The force F₁ it has a magnitude of 70N and is being applied horizontally. The force F_two has a magnitude of 40N and is being applied at an angle of 30 ° to the horizontal plane.

To solve this exercise, a free body diagram is drawn with the coordinate axes x and Y

All components are determined x and Y of the forces acting on the body. The force F₁ has only one horizontal component on the axis x. The force F_two it has two components F_2x  and F_{2 and} which are obtained from the sine and cosine functions of the angle 30 °.

F_1x = F₁=70N

F_2x = F_two cos 30 ° = 40 N.cos 30 ° = 34.64N

F_1y = 0

F_{2 and}= F_two without 30 ° = 40 without 30 ° = 20N

∑ F_x =70N + 34.64N = 104.64N

∑ F_Y=20N + 0 = 20N

Once the resulting forces in the shaft have been determined x and Y we proceed to obtain the numerical value of the resultant force.

F_R^two= (∑ F_x)^two+(∑ F_Y)^two

The resultant force is the square root of the sum of the squared components of the forces

F_R= √ (104.64N)^two+(20N)^two

F_R= 106.53N

The angle formed by the resultant force F_R is obtained from the following expression:

θ_x= so^-1(∑ F_Y / ∑ F_x)

θ_x= so^-1(20N / 104.64N) = 10.82 °

The resultant force F_R has a magnitude of 106.53N and has a direction determined by the angle of 10.82 ° that it forms with the horizontal.

References

1. Dola, G, Duffy, M and Percival, A. Physics. Spain: Heinemann, 2003.
2. Avison, J H. The world of Physics. India: Thomas Nelson and Sons, 1989.
3. Pinsent, M. Physical Processes. United Kingdom: Nelson Thomas, 2002.
4. Yadav, S K. Engineering Mechanics. Delhi: Discovery Publishing House, 2006.
5. Serway, R A and Jewett, J W. Physics for Scientists and Engineers. California, USA: Brooks / Cole, 2010.