# Net force

Last updated

In mechanics, the net force is the vector sum of forces acting on a particle or object. The net force is a single force that replaces the effect of the original forces on the particle's motion. It gives the particle the same acceleration as all those actual forces together as described by Newton's second law of motion.

## Contents

It is possible to determine the torque associated with the point of application of a net force so that it maintains the movement of jets of the object under the original system of forces. Its associated torque, the net force, becomes the resultant force and has the same effect on the rotational motion of the object as all actual forces taken together. [1] It is possible for a system of forces to define a torque-free resultant force. In this case, the net force, when applied at the proper line of action, has the same effect on the body as all of the forces at their points of application. It is not always possible to find a torque-free resultant force.

## Total force

Force is a vector quantity, which means that it has a magnitude and a direction, and it is usually denoted using boldface such as F or by using an arrow over the symbol, such as ${\displaystyle \scriptstyle {\vec {F}}}$.

Graphically, a force is represented as a line segment from its point of application A to a point B, which defines its direction and magnitude. The length of the segment AB represents the magnitude of the force.

Vector calculus was developed in the late 1800s and early 1900s. The parallelogram rule used for the addition of forces, however, dates from antiquity and is noted explicitly by Galileo and Newton. [2]

The diagram shows the addition of the forces ${\displaystyle \scriptstyle {\vec {F}}_{1}}$ and ${\displaystyle \scriptstyle {\vec {F}}_{2}}$. The sum ${\displaystyle \scriptstyle {\vec {F}}}$ of the two forces is drawn as the diagonal of a parallelogram defined by the two forces.

Forces applied to an extended body can have different points of application. Forces are bound vectors and can be added only if they are applied at the same point. The net force obtained from all the forces acting on a body do not preserve its motion unless applied at the same point, and with the appropriate torque associated with the new point of application determined. The net force on a body applied at a single point with the appropriate torque is known as the resultant force and torque.

## Parallelogram rule for the addition of forces

A force is known as a bound vector—which means it has a direction and magnitude and a point of application. A convenient way to define a force is by a line segment from a point A to a point B. If we denote the coordinates of these points as A = (Ax, Ay, Az) and B = (Bx, By, Bz), then the force vector applied at A is given by

${\displaystyle \mathbf {F} =\mathbf {B} -\mathbf {A} =(B_{x}-A_{x},B_{y}-A_{y},B_{z}-A_{z}).}$

The length of the vector B-A defines the magnitude of F and is given by

${\displaystyle |\mathbf {F} |={\sqrt {(B_{x}-A_{x})^{2}+(B_{y}-A_{y})^{2}+(B_{z}-A_{z})^{2}}}.}$

The sum of two forces F1 and F2 applied at A can be computed from the sum of the segments that define them. Let F1 = BA and F2 = DA, then the sum of these two vectors is

${\displaystyle \mathbf {F} =\mathbf {F} _{1}+\mathbf {F} _{2}=\mathbf {B} -\mathbf {A} +\mathbf {D} -\mathbf {A} ,}$

which can be written as

${\displaystyle \mathbf {F} =\mathbf {F} _{1}+\mathbf {F} _{2}=2({\frac {\mathbf {B} +\mathbf {D} }{2}}-\mathbf {A} )=2(\mathbf {E} -\mathbf {A} ),}$

where E is the midpoint of the segment BD that joins the points B and D.

Thus, the sum of the forces F1 and F2 is twice the segment joining A to the midpoint E of the segment joining the endpoints B and D of the two forces. The doubling of this length is easily achieved by defining a segments BC and DC parallel to AD and AB, respectively, to complete the parallelogram ABCD. The diagonal AC of this parallelogram is the sum of the two force vectors. This is known as the parallelogram rule for the addition of forces.

## Translation and rotation due to a force

### Point forces

When a force acts on a particle, it is applied to a single point (the particle volume is negligible): this is a point force and the particle is its application point. But an external force on an extended body (object) can be applied to a number of its constituent particles, i.e. can be "spread" over some volume or surface of the body. However, determining its rotational effect on the body requires that we specify its point of application (actually, the line of application, as explained below). The problem is usually resolved in the following ways:

• Often, the volume or surface on which the force acts is relatively small compared to the size of the body, so that it can be approximated by a point. It is usually not difficult to determine whether the error caused by such approximation is acceptable.
• If it is not acceptable (obviously e.g. in the case of gravitational force), such "volume/surface" force should be described as a system of forces (components), each acting on a single particle, and then the calculation should be done for each of them separately. Such a calculation is typically simplified by the use of differential elements of the body volume/surface, and the integral calculus. In a number of cases, though, it can be shown that such a system of forces may be replaced by a single point force without the actual calculation (as in the case of uniform gravitational force).

In any case, the analysis of the rigid body motion begins with the point force model. And when a force acting on a body is shown graphically, the oriented line segment representing the force is usually drawn so as to "begin" (or "end") at the application point.

### Rigid bodies

In the example shown in the diagram opposite, a single force ${\displaystyle \scriptstyle {\vec {F}}}$ acts at the application point H on a free rigid body. The body has the mass ${\displaystyle \scriptstyle m}$ and its center of mass is the point C. In the constant mass approximation, the force causes changes in the body motion described by the following expressions:

${\displaystyle {\vec {a}}={{\vec {F}} \over m}}$   is the center of mass acceleration; and
${\displaystyle {\vec {\alpha }}={{\vec {\tau }} \over I}}$   is the angular acceleration of the body.

In the second expression, ${\displaystyle \scriptstyle {\vec {\tau }}}$ is the torque or moment of force, whereas ${\displaystyle \scriptstyle I}$ is the moment of inertia of the body. A torque caused by a force ${\displaystyle \scriptstyle {\vec {F}}}$ is a vector quantity defined with respect to some reference point:

${\displaystyle {\vec {\tau }}={\vec {r}}\times {\vec {F}}}$   is the torque vector, and
${\displaystyle \ \tau =Fk}$   is the amount of torque.

The vector ${\displaystyle \scriptstyle {\vec {r}}}$ is the position vector of the force application point, and in this example it is drawn from the center of mass as the reference point of (see diagram). The straight line segment ${\displaystyle \scriptstyle k}$ is the lever arm of the force ${\displaystyle \scriptstyle {\vec {F}}}$ with respect to the center of mass. As the illustration suggests, the torque does not change (the same lever arm) if the application point is moved along the line of the application of the force (dotted black line). More formally, this follows from the properties of the vector product, and shows that rotational effect of the force depends only on the position of its line of application, and not on the particular choice of the point of application along that line.

The torque vector is perpendicular to the plane defined by the force and the vector ${\displaystyle \scriptstyle {\vec {r}}}$, and in this example it is directed towards the observer; the angular acceleration vector has the same direction. The right hand rule relates this direction to the clockwise or counter-clockwise rotation in the plane of the drawing.

The moment of inertia ${\displaystyle \scriptstyle I}$ is calculated with respect to the axis through the center of mass that is parallel with the torque. If the body shown in the illustration is a homogeneous disc, this moment of inertia is ${\displaystyle \scriptstyle I=mr^{2}/2}$. If the disc has the mass 0,5 kg and the radius 0,8 m, the moment of inertia is 0,16 kgm2. If the amount of force is 2 N, and the lever arm 0,6 m, the amount of torque is 1,2 Nm. At the instant shown, the force gives to the disc the angular acceleration α = τ/I = 7,5 rad/s2, and to its center of mass it gives the linear acceleration a = F/m = 4 m/s2.

## Resultant force

Resultant force and torque replaces the effects of a system of forces acting on the movement of a rigid body. An interesting special case is a torque-free resultant, which can be found as follows:

1. Vector addition is used to find the net force;
2. Use the equation to determine the point of application with zero torque:
${\displaystyle {\vec {r}}\times {\vec {F}}_{\mathrm {R} }=\sum _{i=1}^{N}({\vec {r}}_{i}\times {\vec {F}}_{i})}$

where ${\displaystyle {\vec {F}}_{\mathrm {R} }}$ is the net force, ${\displaystyle {\vec {r}}}$ locates its application point, and individual forces are ${\displaystyle {\vec {F}}_{i}}$ with application points ${\displaystyle {\vec {r}}_{i}}$. It may be that there is no point of application that yields a torque-free resultant.

The diagram opposite illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems:

1. Lines of application of the actual forces ${\displaystyle {\vec {F}}_{1}}$ and ${\displaystyle {\vec {F}}_{2}}$ on the leftmost illustration intersect. After vector addition is performed "at the location of ${\displaystyle {\vec {F}}_{1}}$", the net force obtained is translated so that its line of application passes through the common intersection point. With respect to that point all torques are zero, so the torque of the resultant force ${\displaystyle {\vec {F}}_{\mathrm {R} }}$ is equal to the sum of the torques of the actual forces.
2. The illustration in the middle of the diagram shows two parallel actual forces. After vector addition "at the location of ${\displaystyle {\vec {F}}_{2}}$", the net force is translated to the appropriate line of application, where it becomes the resultant force ${\displaystyle \scriptstyle {\vec {F}}_{\mathrm {R} }}$. The procedure is based on decomposition of all forces into components for which the lines of application (pale dotted lines) intersect at one point (the so-called pole, arbitrarily set at the right side of the illustration). Then the arguments from the previous case are applied to the forces and their components to demonstrate the torque relationships.
3. The rightmost illustration shows a couple, two equal but opposite forces for which the amount of the net force is zero, but they produce the net torque ${\displaystyle \scriptstyle \tau =Fd}$   where ${\displaystyle \scriptstyle \ d}$  is the distance between their lines of application. Since there is no resultant force, this torque can be [is?] described as "pure" torque.

## Usage

In general, a system of forces acting on a rigid body can always be replaced by one force plus one pure (see previous section) torque. The force is the net force, but to calculate the additional torque, the net force must be assigned the line of action. The line of action can be selected arbitrarily, but the additional pure torque depends on this choice. In a special case, it is possible to find such line of action that this additional torque is zero.

The resultant force and torque can be determined for any configuration of forces. However, an interesting special case is a torque-free resultant. This is useful, both conceptually and practically, because the body moves without rotating as if it was a particle.

Some authors do not distinguish the resultant force from the net force and use the terms as synonyms. [3]

## Related Research Articles

In physics, angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, frisbees rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N). Force is represented by the symbol F.

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives:

In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. It represents the capability of a force to produce change in the rotational motion of the body. The concept originated with the studies by Archimedes of the usage of levers, which is reflected in his famous quote: "Give me a lever and a place to stand and I will move the Earth". Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Torque is defined as the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity. This is a very good approximation for smaller bodies near the surface of Earth, so there is no practical need to distinguish "center of gravity" from "center of mass" in most applications, such as engineering and medicine.

In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

The parallelogram of forces is a method for solving the results of applying two forces to an object.

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.

In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces. Calculating and visualizing the resultant force on a body is done through computational analysis, or a free body diagram.

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

Dynamical simulation, in computational physics, is the simulation of systems of objects that are free to move, usually in three dimensions according to Newton's laws of dynamics, or approximations thereof. Dynamical simulation is used in computer animation to assist animators to produce realistic motion, in industrial design, and in video games. Body movement is calculated using time integration methods.

In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms.

Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

In physics, the line of action of a force (F) is a geometric representation of how the force is applied. It is the line through the point at which the force is applied in the same direction as the vector F.

In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. The diagram shows a beam which is simply supported at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed ; therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.

In mechanics, a couple is a system of forces with a resultant moment but no resultant force.

In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.

## References

1. Symon, Keith R. (1964), Mechanics, Addison-Wesley, LCCN   60-5164
2. Michael J. Crowe (1967). A History of Vector Analysis : The Evolution of the Idea of a Vectorial System. Dover Publications (reprint edition; ISBN   0-486-67910-1).
3. Resnick, Robert and Halliday, David (1966), Physics, (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527