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**2.1 Forces**

In this chapter we review the basic concepts of forces, and force laws. Most of this material is identical to material covered in EN030, and is provided here as a review. There are a few additional sections â€“ for example forces exerted by a damper or dashpot, an inerter, and interatomic forces are discussed in Section 2.1.7.

**2.1.1 Definition of a force **

Engineering design calculations nearly always use classical (Newtonian) mechanics. In classical mechanics, the concept of a `forceâ€™ is based on experimental observations that everything in the universe seems to have a preferred configuration â€“ masses appear to attract each other; objects with opposite charges attract one another; magnets can repel or attract one another; you are probably repelled by your professor. But we donâ€™t really know why this is (except perhaps the last one).

The idea of a force is introduced to quantify the tendency of objects to move towards their preferred configuration. If objects accelerate very quickly towards their preferred configuration, then we say that thereâ€™s a big force acting on them. If they donâ€™t move (or move at constant velocity), then there is no force. We canâ€™t see a force; we can only deduce its existence by observing its effect.

Specifically, forces are defined through Newtonâ€™s laws of motion

0. A `particleâ€™ is a small mass at some position in space.

1. When the sum of the forces acting on a particle is zero, its velocity is constant;

2. The sum of forces acting on a particle of constant mass is equal to the product of the mass of the particle and its acceleration;

3. The forces exerted by two particles on each other are equal in magnitude and opposite in direction.

The second law provides the definition of a force â€“ if a mass m has acceleration a, the force F acting on it is

**F = ma**

Of course, there is a big problem with Newtonâ€™s laws â€“ what do we take as a fixed point (and orientation) in order to define acceleration? The general theory of relativity addresses this issue rigorously. But for engineering calculations we can usually take the earth to be fixed, and happily apply Newtonâ€™s laws. In rare cases where the earthâ€™s motion is important, we take the stars far from the solar system to be fixed.

**2.1.2 Causes of force**

Forces may arise from a number of different effects, including

(i) Gravity;

(ii) Electromagnetism or electrostatics;

(iii) Pressure exerted by fluid or gas on part of a structure

(v) Wind or fluid induced drag or lift forces;

(vi) Contact forces, which act wherever a structure or component touches anything;

(vii) Friction forces, which also act at contacts.

Some of these forces can be described by universal laws. For example, gravity forces can be calculated using Newtonâ€™s law of gravitation; electrostatic forces acting between charged particles are governed by Coulombâ€™s law; electromagnetic forces acting between current carrying wires are governed by Ampereâ€™s law; buoyancy forces are governed by laws describing hydrostatic forces in fluids. Some of these universal force laws are listed in Section 2.6.

Some forces have to be measured. For example, to determine friction forces acting in a machine, you may need to measure the coefficient of friction for the contacting surfaces. Similarly, to determine aerodynamic lift or drag forces acting on a structure, you would probably need to measure its lift and drag coefficient experimentally. Lift and drag forces are described in Section 2.6. Friction forces are discussed in Section 12.

Contact forces are pressures that act on the small area of contact between two objects. Contact forces can either be measured, or they can be calculated by analyzing forces and deformation in the system of interest. Contact forces are very complicated, and are discussed in more detail in Section 8.

**2.1.3 Units of force and typical magnitudes**

In SI units, the standard unit of force is the Newton, given the symbol N.

The Newton is a derived unit, defined through Newtonâ€™s second law of motion â€“ a force of 1N causes a 1 kg mass to accelerate at 1 ms^{âˆ’2}.

The fundamental unit of force in the SI convention is kg m/s^{2}

In US units, the standard unit of force is the pound, given the symbol lb or lbf (the latter is an abbreviation for pound force, to distinguish it from pounds weight)

A force of 1 lbf causes a mass of 1 slug to accelerate at 1 ft/s^{2}

US units have a frightfully confusing way of representing mass â€“ often the mass of an object is reported as weight, in lb or lbm (the latter is an abbreviation for pound mass). The weight of an object in lb is not mass at all â€“ itâ€™s actually the gravitational force acting on the mass. Therefore, the mass of an object in slugs must be computed from its weight in pounds using the formula

where g=32.1740 ft/s^{2} is the acceleration due to gravity.

A force of 1 lb(f) causes a mass of 1 lb(m) to accelerate at 32.1740 ft/s^{2}

The conversion factors from lb to N are

1 lb | = | 4.448 N |

1 N | = | 0.2248 lb |

As a rough guide, a force of 1N is about equal to the weight of a medium sized apple. A few typical force magnitudes (from `The Sizesaurusâ€™, by Stephen Strauss, Avon Books, NY, 1997) are listed in the table below

Force | Newtons | Pounds Force |

Gravitational Pull of the Sun on Earth | 3.5 x10 | 7.9 x10 |

Gravitational Pull of the Earth on the Moon | 2 x10 | 4.5 x10 |

Thrust of a Saturn V rocket engine | 3.3 x10 | 7.4 x10 |

Thrust of a large jet engine | 7.7 x10 | 1.7 x10 |

Pull of a large locomotive | 5 x10 | 1.1x10 |

Force between two protons in a nucleus | 10 | 10 |

Gravitational pull of the earth on a person | 7.3 x10 | 1.6 x10 |

Maximum force exerted upwards by a forearm | 2.7 x10 | 60 |

Gravitational pull of the earth on a 5 cent coin | 5.1x10 | 1.1 x10 |

Force between an electron and the nucleus of a Hydrogen atom | 8 x10 | 1.8 x10 |

**2.1.4 Classification of forces: External forces, constraint forces and internal forces.**

When analyzing forces in a structure or machine, it is conventional to classify forces as external forces; constraint forces or internal forces.

External forces arise from interaction between the system of interest and its surroundings.

Examples of external forces include gravitational forces; lift or drag forces arising from wind loading; electrostatic and electromagnetic forces; and buoyancy forces; among others. Force laws governing these effects are listed later in this section.

Constraint forces are exerted by one part of a structure on another, through joints, connections or contacts between components. Constraint forces are very complex, and will be discussed in detail in Section 8.

Internal forces are forces that act inside a solid part of a structure or component. For example, a stretched rope has a tension force acting inside it, holding the rope together. Most solid objects contain very complex distributions of internal force. These internal forces ultimately lead to structural failure, and also cause the structure to deform. The purpose of calculating forces in a structure or component is usually to deduce the internal forces, so as to be able to design stiff, lightweight and strong components. We will not, unfortunately, be able to develop a full theory of internal forces in this course â€“ a proper discussion requires understanding of partial differential equations, as well as vector and tensor calculus. However, a brief discussion of internal forces in slender members will be provided in Section 9.

**2.1.5 Mathematical representation of a force.**

Force is a vector â€“ it has a magnitude (specified in Newtons, or lbf, or whatever), and a direction.

A force is therefore always expressed mathematically as a vector quantity. To do so, we follow the usual rules, which are described in more detail in the vector tutorial. The procedure is

1. Choose basis vectors **{i, j, k} or {e _{1}, e_{2}, e_{2}}** that establish three fixed (and usually perpendicular) directions in space;

2. Using geometry or trigonometry, calculate the force component along each of the three reference directions (F_{x}, F_{y}, F_{z}) or (F_{1}, F_{2}, F_{3}) ;

3. The vector force is then reported as

**F = F _{x} i + F_{y} j + F_{z} k = F_{1}e_{1} + F_{2}e_{2} + F_{3}e_{3} (appropriate units)**

For calculations, you will also need to specify the point where the force acts on your system or structure. To do this, you need to report the position vector of the point where the force acts on the structure.

The procedure for representing a position vector is also described in detail in the vector tutorial. To do so, you need to:

1. Choose an origin

2. Choose basis vectors **{i, j, k} or {e _{1}, e_{2}, e_{2}} **that establish three fixed directions in space (usually we use the same basis for both force and position vectors) (rx, ry,rz) or (r

4. The position vector is then reported as

**r = r _{x}i + r_{y}j + r_{z}k = r_{1}e_{1}+ r_{2}e_{2} + r_{3},e_{3} (appropriate units)**

**2.1.6 Measuring forces**

Engineers often need to measure forces. According to the definition, if we want to measure a force, we need to get hold of a 1 kg mass, have the force act on it somehow, and then measure the acceleration of the mass. The magnitude of the acceleration tells us the magnitude of the force; the direction of motion of the mass tells us the direction of the force. Fortunately, there are easier ways to measure forces.

In addition to causing acceleration, forces cause objects to deform â€“ for example, a force will stretch or compress a spring; or bend a beam. The deformation can be measured, and the force can be deduced.

The simplest application of this phenomenon is a spring scale. The change in length of a spring is proportional to the magnitude of the force causing it to stretch (so long as the force is not too large!)â€“ this relationship is known as Hookeâ€™s law and can be expressed as an equation

kÎ´ = F

where the spring stiffness k depends on the material the spring is made from, and the shape of the spring. The spring stiffness can be measured experimentally to calibrate the spring.

Spring scales are not exactly precision instruments, of course. But the same principle is used in more sophisticated instruments too. Forces can be measured precisely using a `force transducerâ€™ or `load cellâ€™ (A search for `force transducerâ€™ on any search engine will bring up a huge variety of these â€“ a few are shown in the picture). The simplest load cell works much like a spring scale â€“ you can load it in one direction, and it will provide an electrical signal proportional to the magnitude of the force. Sophisticated load cells can measure a force vector, and will record all three force components. Really fancy load cells measure both force vectors, and torque or moment vectors.

Simple force transducers capable of measuring a single force component. The instrument on the right is called a `proving ringâ€™ â€“ thereâ€™s a short article describing how it works at

A sophisticated force transducer produced by MTS systems, which is capable of measuring forces and moments acting on a carâ€™s wheel in-situ.

The basic design of all these load cells is the same â€“ they measure (very precisely) the deformation in a part of the cell that acts like a very stiff spring.

In this case the `springâ€™ is actually a tubular piece of high-strength steel. When a force acts on the cylinder, its length decreases slightly. The deformation is detected using `strain gagesâ€™ attached to the cylinder. A strain gage is really just a thin piece of wire, which deforms with the cylinder. When the wire gets shorter, its electrical resistance decreases â€“ this resistance change can be measured, and can be used to work out the force. It is possible to derive a formula relating the force to the change in resistance, the load cell geometry, and the material properties of steel, but the calculations involved are well beyond the scope of this course.

The most sensitive load cell currently available is the atomic force microscope (AFM) â€“ which as the name suggests, is intended to measure forces between small numbers of atoms. This device consists of a very thin (about 1 Î¼m) cantilever beam, clamped at one end, with a sharp tip mounted at the other. When the tip is brought near a sample, atomic interactions exert a force on the tip and cause the cantilever to bend. The bending is detected by a laser-mirror system. The device is capable of measuring forces of about 1 pN (thatâ€™s 10^{âˆ’12} N!!), and is used to explore the properties of surfaces, and biological materials such as DNA strands and cell membranes.

Selecting a load cell

As an engineer, you may need to purchase a load cell to measure a force. Here are a few considerations that will guide your purchase.

1. How many force (and maybe moment) components do you need to measure? Instruments that measure several force components are more expensiveâ€¦

2. Load capacity â€“ what is the maximum force you need to measure?

3. Load range â€“ what is the minimum force you need to measure?

4. Accuracy

5. Temperature stability â€“ how much will the reading on the cell change if the temperature changes?

6. Creep stability â€“ if a load is applied to the cell for a long time, does the reading drift?

7. Frequency response â€“ how rapidly will the cell respond to time varying loads? What is the maximum frequency of loading that can be measured?

8. Reliability

9. Cost

**2.1.7 Force Laws**

In this section, we list equations that can be used to calculate forces associated with

(i) Gravity

(ii) Forces exerted by linear springs

(iii) Electrostatic forces

(iv) Electromagnetic forces

(v) Hydrostatic forces and buoyancy

(vi) Aero- and hydro-dynamic lift and drag forces

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