Statics

 

General Principles

You can find the preliminary information required for the static course on this blog. General information about mechanics, the difference between static and dynamics, basic concepts, units of measure, International System of Units and Numerical Calculations.

Mechanics

Mechanics is a branch of the physical sciences that are concerned with the state of rest or motion of bodies that are subjected to the action of forces. In general, this subject can be subdivided into three branches: rigid-body mechanics, deformable-body mechanics, and fluid mechanics. Rigid-body mechanics is essential for the design and analysis of many types of structural members, mechanical components, or electrical devices encountered in engineering. Rigid-body mechanics is divided into two areas: statics and dynamics. Statics deals with the equilibrium of bodies, that is, those that are either at rest or move with a constant velocity; whereas dynamics is concerned with the accelerated motion of bodies.

 

Fundamental Concepts

Basic Quantities. The following four quantities are used throughout mechanics.

Length is used to locate the position of a point in space and thereby describe the size of a physical system. Once a standard unit of length is defined, one can then use it to define distances and geometric properties of a body as multiples of this unit.

Time is conceived as a succession of events. Although the principles of statics are time-independent, this quantity plays an important role in the study of dynamics.

Mass is a measure of a quantity of matter that is used to compare the action of one body with that of another. This property manifests itself as a gravitational attraction between two bodies and provides a measure of the resistance of matter to a change in velocity.

Force. In general, force is considered as a “push” or “pull” exerted by one body on another. This interaction can occur when there is direct contact between the bodies, such as a person pushing on a wall, or it can occur through a distance when the bodies are physically separated.

Examples of the latter type include gravitational, electrical, and magnetic forces. In any case, a force is completely characterized by its magnitude, direction, and point of application.

 

Newton’s Three Laws of Motion. Engineering mechanics is formulated based on Newton’s three laws of motion, the validity of which is based on experimental observation. These laws apply to the motion of a particle as measured from a nonaccelerating reference frame. They may be briefly stated as follows.

 

First Law. A particle originally at rest, or moving in a straight line with constant velocity, tends to remain in this state provided the particle is not subjected to an unbalanced force

Second Law. A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force, If F is applied to a particle of mass m, this law may be expressed mathematically as

F = ma

Third Law. The mutual forces of action and reaction between two particles are equal, opposite, and collinear,

Newton’s Law of Gravitational Attraction. Shortly after formulating his three laws of motion, Newton postulated a law governing the gravitational attraction between any two particles. Stated mathematically

Weight. According to Eq. 1–2, any two particles or bodies have a mutual attractive (gravitational) force acting between them. In the case of a particle located at or near the surface of the earth, however, the only gravitational force having any sizable magnitude is that between the earth and the particle. Consequently, this force, termed the weight, will be the only gravitational force considered in the study of mechanics.

W = mg

Units of Measurement

The four basic quantities—length, time, mass, and force—are not all independent from one another; in fact, they are related by Newton’s second law of motion, F = ma. Because of this, the units used to measure these quantities cannot all be selected arbitrarily. The equality F = ma is maintained only if three of the four units called base units, are defined and the fourth unit is then derived from the equation.

 

SI Units

U.S. Units

 

The International System of Units

When a numerical quantity is either very large or very small, the units used to define its size may be modified by using a prefix. For example, 4 000 000 N = 4 000 kN (kilo-newton) = 4 MN (mega-newton), or 0.005 m = 5 mm (millimeter). Except for some volume and area measurements, the use of these prefixes is to be avoided in science and engineering.

Notice that the SI the system does not include the multiple deca (10) or the submultiple centi (0.01), which form part of the metric system.

Rules for Use. Here are a few of the important rules that describe the proper use of the various SI symbols:

• Quantities defined by several units which are multiples of one another are separated by a dot to avoid confusion with prefix notation, as indicated by N = kg. m/s² = kg . m. s^-2. Also, m. s

(meter-second), whereas ms (millisecond).

• The exponential power on a unit having a prefix refers to both the unit and its prefix. For example, mN² = (mN)² = mN . mN. Likewise, mm2 represents (mm)² = mm . mm.

• except for the base unit the kilogram, in general, avoids the use of a prefix in the denominator of composite units. For example, do not write N>mm, but rather kN>m; also, m>mg should be written as Mm>kg.

• When performing calculations, represent the numbers in terms of their base or derived units by converting all prefixes to powers of 10. The final result should then be expressed using a single prefix. Also, after calculation, it is best to keep numerical values between 0.1 and

1000; otherwise, a suitable prefix should be chosen. For example, (50 kN) (60 nm) = [50(10³) N] [60(10^-9) m] = 3000(10^-6) N. m = 3(10^-3) N . m = 3 mN . m

 

Numerical Calculations

The terms of any equation used to describe a physical process must be dimensionally homogeneous; that is, each term must be expressed in the same units. Provided this is the case, all the terms of an equation can then be combined if numerical values

are substituted for the variables.

s = vt + 1/2 at² à [m, (m>s)s, (m>s²)s²]

 

The number of significant figures contained in any number determines the accuracy of the number. For instance, the number 4981 contains four significant figures. However, if zeros occur at the end of a whole number, it may be unclear as to how many significant figures the number represents. For example, 23 400 might have three (234), four (2340), or five (23 400) significant figures. To avoid these ambiguities, we will use engineering notation to report a result. This requires that numbers be rounded off to the appropriate number of significant digits and then expressed in multiples of (10³), such as (10³), (10⁶), or (10^-9). For instance, if 23 400 has five significant figures, it is written as 23.400(10³), but if it has only three significant figures, it is written as 23.4(10³). If zeros occur at the beginning of a number that is less than one, then the zeros are not significant. For example, 0.008 21 has three significant figures. Using engineering notation, this number is expressed as 8.21(10^-3). Likewise, 0.000 582 can be expressed as 0.582(10^-3) or 582(10^-6).

 

Rounding off a number is necessary so that the accuracy of the result will be the same as that of the problem data. As a general rule, any numerical figure ending in a number greater than five is rounded up and a number less than five is not rounded up. The rules for rounding off numbers are best illustrated by examples. Suppose the number 3.5587 is to be rounded off to three significant figures. Because the fourth digit (8) is greater than 5, the third number is rounded up to 3.56. Likewise, 0.5896 becomes 0.590 and 9.3866 becomes 9.39. If we round off 1.341 to three significant figures, because the fourth digit (1) is less than 5, then we get 1.34. Likewise 0.3762 becomes 0.376 and 9.871 becomes 9.87. There is a special case for any number that ends in a 5. As a general rule, if the digit preceding the 5 is an even number, then this digit is not rounded up. If the digit preceding the 5 is an odd number, then it is rounded up. For example, 75.25 rounded off to three significant digits becomes 75.2, 0.1275 becomes 0.128, and 0.2555 becomes 0.256.

References:

https://en.wikipedia.org/wiki/Statics#:~:text=Statics%20is%20the%20branch%20of,static%20equilibrium%20with%20their%20environment.

https://www.britannica.com/science/statics

http://civilittee-hu.com/uploads/1/Static/book14th.pdf