Free body diagram

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Block on a ramp and corresponding free body diagram of the block. Free body diagram2.svg
Block on a ramp and corresponding free body diagram of the block.

In physics and engineering, a free body diagram (FBD; also called a force diagram) [1] is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a free body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems. Sometimes in order to calculate the resultant force graphically the applied forces are arranged as the edges of a polygon of forces [2] or force polygon (see § Polygon of forces).

Contents

Free body

A body is said to be "free" when it is singled out from other bodies for the purposes of dynamic or static analysis. The object does not have to be "free" in the sense of being unforced, and it may or may not be in a state of equilibrium; rather, it is not fixed in place and is thus "free" to move in response to forces and torques it may experience.

Figure 1: The red cylinder is the "free" body, the body of interest. Red cylinder freed.PNG
Figure 1: The red cylinder is the "free" body, the body of interest.
Figure 2: Now the left half of the cylinder is the "free" body. Left half of red cylinder freed.PNG
Figure 2: Now the left half of the cylinder is the "free" body.

Figure 1 shows, on the left, green, red, and blue widgets stacked on top of each other, and for some reason the red cylinder happens to be the body of interest. (It may be necessary to calculate the stress to which it is subjected, for example.) On the right, the red cylinder has become the free body. In figure 2, the interest has shifted to just the left half of the red cylinder and so now it is the free body on the right. The example illustrates the context sensitivity of the term "free body". A cylinder can be part of a free body, it can be a free body by itself, and, as it is composed of parts, any of those parts may be a free body in itself. Figure 1 and 2 are not yet free body diagrams. In a completed free body diagram, the free body would be shown with forces acting on it. [3]

Purpose

Free body diagrams are used to visualize forces and moments applied to a body and to calculate reactions in mechanics problems. These diagrams are frequently used both to determine the loading of individual structural components and to calculate internal forces within a structure. They are used by most engineering disciplines from Biomechanics to Structural Engineering. [4] [5] In the educational environment, a free body diagram is an important step in understanding certain topics, such as statics, dynamics and other forms of classical mechanics.

Features

A free body diagram is not a scaled drawing, it is a diagram. The symbols used in a free body diagram depends upon how a body is modeled. [6]

Free body diagrams consist of:

The number of forces and moments shown depends upon the specific problem and the assumptions made. Common assumptions are neglecting air resistance and friction and assuming rigid body action.

In statics all forces and moments must balance to zero; the physical interpretation is that if they do not, the body is accelerating and the principles of statics do not apply. In dynamics the resultant forces and moments can be non-zero.

Free body diagrams may not represent an entire physical body. Portions of a body can be selected for analysis. This technique allows calculation of internal forces, making them appear external, allowing analysis. This can be used multiple times to calculate internal forces at different locations within a physical body.

For example, a gymnast performing the iron cross: modeling the ropes and person allows calculation of overall forces (body weight, neglecting rope weight, breezes, buoyancy, electrostatics, relativity, rotation of the earth, etc.). Then remove the person and show only one rope; you get force direction. Then only looking at the person the forces on the hand can be calculated. Now only look at the arm to calculate the forces and moments at the shoulders, and so on until the component you need to analyze can be calculated.

Modeling the body

A body may be modeled in three ways:

What is included

An FBD represents the body of interest and the external forces acting on it.

Often a provisional free body is drawn before everything is known. The purpose of the diagram is to help to determine magnitude, direction, and point of application of external loads. When a force is originally drawn, its length may not indicate the magnitude. Its line may not correspond to the exact line of action. Even its orientation may not be correct.

External forces known to have negligible effect on the analysis may be omitted after careful consideration (e.g. buoyancy forces of the air in the analysis of a chair, or atmospheric pressure on the analysis of a frying pan).

External forces acting on an object may include friction, gravity, normal force, drag, tension, or a human force due to pushing or pulling. When in a non-inertial reference frame (see coordinate system, below), fictitious forces, such as centrifugal pseudoforce are appropriate.

At least one coordinate system is always included, and chosen for convenience. Judicious selection of a coordinate system can make defining the vectors simpler when writing the equations of motion or statics. The x direction may be chosen to point down the ramp in an inclined plane problem, for example. In that case the friction force only has an x component, and the normal force only has a y component. The force of gravity would then have components in both the x and y directions: mgsin(θ) in the x and mgcos(θ) in the y, where θ is the angle between the ramp and the horizontal.

Exclusions

A free body diagram should not show:

Analysis

In an analysis, a free body diagram is used by summing all forces and moments (often accomplished along or about each of the axes). When the sum of all forces and moments is zero, the body is at rest or moving and/or rotating at a constant velocity, by Newton's first law. If the sum is not zero, then the body is accelerating in a direction or about an axis according to Newton's second law.

Forces not aligned to an axis

Angled force (F) redefined into components along axes (Fx) and (Fy) Vector components.JPG
Angled force (F) redefined into components along axes (Fx) and (Fy)

Determining the sum of the forces and moments is straightforward if they are aligned with coordinate axes, but it is more complex if some are not. It is convenient to use the components of the forces, in which case the symbols ΣFx and ΣFy are used instead of ΣF (the variable M is used for moments).

Forces and moments that are at an angle to a coordinate axis can be rewritten as two vectors that are equivalent to the original (or three, for three dimensional problems)—each vector directed along one of the axes (Fx) and (Fy).

Example: A block on an inclined plane

A simple free-body diagram, shown above, of a block on a ramp, illustrates this.

Some care is needed in interpreting the diagram.

Polygon of forces

A force polygon for the forces P1 to P6 applied to point O Polygon of forces.png
A force polygon for the forces P1 to P6 applied to point O

In the case of two applied forces, their sum (resultant force) can be found graphically using a parallelogram of forces.

To graphically determine the resultant force of multiple forces, the acting forces can be arranged as edges of a polygon by attaching the beginning of one force vector to the end of another in an arbitrary order. Then the vector value of the resultant force would be determined by the missing edge of the polygon. [2] In the diagram, the forces P1 to P6 are applied to the point O. The polygon is constructed starting with P1 and P2 using the parallelogram of forces (vertex a). The process is repeated (adding P3 yields the vertex b, etc.). The remaining edge of the polygon O-e represents the resultant force R.

Kinetic diagram

Free body and kinetic diagrams of an inclined block Kinetic diagram of inclined block.svg
Free body and kinetic diagrams of an inclined block

In dynamics a kinetic diagram is a pictorial device used in analyzing mechanics problems when there is determined to be a net force and/or moment acting on a body. They are related to and often used with free body diagrams, but depict only the net force and moment rather than all of the forces being considered.

Kinetic diagrams are not required to solve dynamics problems; their use in teaching dynamics is argued against by some [7] in favor of other methods that they view as simpler. They appear in some dynamics texts [8] but are absent in others. [9]

See also

Related Research Articles

<span class="mw-page-title-main">Force</span> Influence that can change motion of an object

A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.

<span class="mw-page-title-main">Friction</span> Force resisting sliding motion

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of the processes involved is called tribology, and has a history of more than 2000 years.

<span class="mw-page-title-main">Precession</span> Periodic change in the direction of a rotation axis

Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment.

<span class="mw-page-title-main">Work (physics)</span> Process of energy transfer to an object via force application through displacement

In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. 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.

<span class="mw-page-title-main">Center of mass</span> Unique point where the weighted relative position of the distributed mass sums to zero

In physics, the center of mass of a distribution of mass in space is the unique point at any given time 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.

<span class="mw-page-title-main">Truss</span> Rigid structure that consists of two-force members only

A truss is an assembly of members such as beams, connected by nodes, that creates a rigid structure.

<span class="mw-page-title-main">Net force</span> Vector sum of all forces acting upon a particle or body

In mechanics, the net force is the sum of all the forces acting on an object. For example, if two forces are acting upon an object in opposite directions, and one force is greater than the other, the forces can be replaced with a single force that is the difference of the greater and smaller force. That force is the net force.

<span class="mw-page-title-main">Rigid body</span> Physical object which does not deform when forces or moments are exerted on it

In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible. 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.

<span class="mw-page-title-main">Resultant force</span> Single force representing the combination of all forces acting on a physical body

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The work of a force on a particle along a virtual displacement is known as the virtual work.

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<span class="mw-page-title-main">Bending moment</span> Force tending to bend a structural element

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References

  1. "Force Diagrams (Free-body Diagrams)". Western Kentucky University. Archived from the original on 2011-03-17. Retrieved 2011-03-17.
  2. 1 2 Rennie & Law 2019.
  3. Ellse, Mark; Honeywell, Chris (1997). Mechanics and Electricity.
  4. Ruina, Andy; Pratap, Rudra (2010). Introduction to Statics and Dynamics (PDF). Oxford University Press. pp. 79–105. Retrieved 2006-08-04.
  5. Hibbeler, R.C. (2007). Engineering Mechanics: Statics & Dynamics (11th ed.). Pearson Prentice Hall. pp. 83–86. ISBN   978-0-13-221509-1.
  6. Puri, Avinash (1996). "The Art of Free-body Diagrams". Physics Education. 31 (3): 155. Bibcode:1996PhyEd..31..155P. doi:10.1088/0031-9120/31/3/015. S2CID   250802652.
  7. Kraige, L. Glenn (16 June 2002), The Role Of The Kinetic Diagram In The Teaching Of Introductory Rigid Body Dynamics Past, Present, And Future, pp. 7.1182.1–7.1182.11
  8. "Stress and Dynamics" (PDF). Retrieved August 5, 2015.
  9. Ruina, Andy; Pratap, Rudra (2002). Introduction to Statics and Dynamics. Oxford University Press. Retrieved September 4, 2019.

Sources

Notes

  1. The line of action is important where moment matters