Introduction

1. Introduction Concepts, Principles, Units

2. What is Statics • that branch of Mechanics which deals with the study of the conditions under which rigid bodies are at rest, i.e. in a state of static equilibrium. Mechanics (derived from a Greek word meaning "contrivances" i.e. gadgets, mechanical devices, contraptions) is that branch of the applied sciences that describes and predicts the conditions of rest or motion of bodies that are subjected to the action of forces.

4. Basic Concepts • Space • Time • Mass • Force

5. Space • The geometric region within which a body or particle may be located. • The position of a point in space may be described by polar co-ordinates (linear and angular measurements relative to a co-ordinate system) or by specifying rectangular co-ordinates, i.e. three independent co-ordinates of the point. • The concept of length is the quantity this is used to locate the position of a point in space as well as to describe the size of a physical system.

6. Time • Is that quantity that specifies/positions the occurrence of an event relative to other events. • While the principles of statics are independent of time, time is a basic quantity of great importance in the study of dynamics. Mass • May be defined as the quantity of matter in a body. It provides a quantitative measure of the resistance of a body to a change in velocity, i.e. it is a measure of the inertia of a body.

7. Force • That effect which tends to cause a body to change its state of rest or uniform motion. • Represents the action of one body on another and may be excited by actual contact or at a distance (e.g. gravitational, electrical and magnetic forces). • Force is a vector quantity and hence is completely characterized by its point of application, its magnitude and its direction.

8. In Newtonian mechanics (as opposed to relativistic mechanics), the three basic quantities of length, time and mass are independent of each other. The fourth concept, force, is related to the other two by Newton’s 2nd Law of Motion which can be stated mathematically as F = ma

9. Fundamental Principles of Mechanics • Newton's First Law • Newton's Second Law • Newton's Third Law • Newton’s Law of Gravitational Attraction • Parallelogram law for the Addition of Two Forces • Principle of Transmissibility

10. Fundamental Principles Contd... • First Law: If the resultant force acting on a particle is zero, the particle will remain at rest (if originally at rest) or will move with a constant speed in a straight line (if originally in motion). • Second Law: If the resultant force acting on a particle is not zero, the particle will have an acceleration which is proportional to the magnitude of the resultant force and in the direction of this force, i.e. F = ma • Third Law: The forces of action and reaction between bodies in contact have the same magnitude, same line of action but opposite sense. • Newton’s 3rd Law is applied when considering the equilibrium of rigid bodies, where it may be necessary to dismember an object to completely analyse it.

11. Fundamental Principles Contd... Newton's Law of Gravitational Attraction: • two particles of mass M and m are mutually attracted to each other with equal and opposite forces F and -F, where the magnitude of F is given by the expression F = G Mm/r2 where G = universal constant of gravitation = 66.73 x 10-12 m3/(kgs2) = 66.73 x 10-12 Nm2/kg2) r = distance between the two particles Newton’s law of gravitation attraction is used to obtain the acceleration due to gravity, g of any object which is located on or near the earth’s surface.

12. Fundamental Principles Contd... • Parallelogram law for the Addition of Two Forces: states that two forces acting on a particle may be replaced by a single force, called the resultant force which is equivalent to both forces and is obtained by drawing the diagonal of the parallelogram which has sides equal to the given forces. • This principle can be applied when analysing a system of coplanar forces which is acting on a particle.

13. Fundamental Principles Contd... • Principle of Transmissibility: This states that the conditions of equilibrium or of motion of a rigid body will remain unchanged if a force acting at a given point on the rigid body is replaced by a force of the same magnitude and same direction, but acting at a different point, provided that the two forces have the same line of action. • This Principle is of importance in the study of the statics of rigid body.

14. Idealization • Involves the reduction of a problem from its physical description to a model to which the principles of mechanics can be applied. This is done to simplify the problem such that the principles of mechanics may be applied to it. Examples: • Particle (point mass): is a body whose size is negligible in comparison to the over all dimensions of the problem being analyzed. It is a small amount of matter and is assumed to occupy a single point in space • Rigid Body: is a body in which the deformation occurring as a result of the forces applied on it is relatively small, i.e. negligible.

15. Units of Measurement • Kinetic Units: units used to define the basic quantities of force, mass, length and time Note that the quantities associated with the basic quantities of length, time, mass and force are not all independent from one another but are related by Newton’s 2nd Law of motion, which expressed mathematically states that F = ma Equality is maintained if three of the four units, called base units, are arbitrarily defined while the fourth is derived from the equation and is referred to as a derived unit. Units of measurements that are so selected that dimensional homogeneity is maintained are said to form a consistent system of units.

16. SI units system (Systeme International d' Unites) • base units are the units of length(m), mass(kg) and time(s) • unit of force is a derived unit called the newton (N) • defined as the force which gives an acceleration of 1 m/s2 to a mass of 1 kg • multiples and submultiples of the fundamental SI units and the derived units are obtained through the use of prefixes like kilo-, mega-, etc.

17. U.S. Customary system of Units • base units are length (ft), time (s) and force (lb). • The unit of mass is a derived is called the slug • one slug is the quantity of matter that is accelerated by 1ft/s2 when acted upon by a force of one pound (1 lb)

18. slug Summary of Units Angular Units: Degrees Radians

19. Accuracy of Numerical Calculations Significant Figures: number of precise digits in a number, determined by the least accurate of all the data involved in a given problem • The number of significant figures is determined by starting from the first non-zero digit and counting to the right • 7.340 & 0.007340 both have 4 sig. figures

20. Accuracy of Numerical Calculations • mathematical formulation of a physical problem often represents an ideal limiting description, or model • idealized mathematical model involves certain assumptions/approximations • approximations may be physical or mathematical

21. Accuracy of Numerical Calculations • dimensionally homogeneous - each term in any equation used to describe a physical system must be expressed in the same set of consistent units • accuracy of a solution can never be greater than that of the problem data

22. Rules for Rounding ‘N’ Sig. Digits • If the n+1 digit is less than 5, then the n+1 digit and others following it are dropped. e.g. 2.326 and 0.451, rounded off to n=2 significant figures, would be 2.3 and 0.45. • If the n+1 digit is equal to 5 with zeros following it, then round off the nth digit to an even number, e.g. 1245 and 0.86550 rounded off to 3 significant figures become 1240 and 0.866

23. Rules for Rounding ‘N’ Sig. Digits • If the n+1 digit is equal to or greater than 5 with non zero digits following it, then increase the nth digit by 1 and drop the n+1 digit and the others following it. e.g. 0.72387 and 565.5003 rounded off to 3 significant figures become 0.724 and 566. • As a general rule, always retain one extra significant figure in your computation than in the least accurate data in the problem. Then round off your final result so that it has the same number of significant figures as the least accurate number

24. General Procedure for Performing Calculations Present the work in a logical, orderly, and neat manner, as suggested by the following sequence of steps: • Read the problem carefully. • Reduce the problem from its physical description to a model to which the principles of mechanics can be applied • try to correlate the actual physical situation with the theory studied. • State the given data and state the results required. Draw the necessary free body diagrams.

25. General Procedure for Performing Calculations • Apply the relevant principles of mechanics, generally in mathematical form. • Solve the necessary equations algebraically as far as practical, then, making sure they are dimensionally homogeneous, use a consistent set of units and complete the solution numerically. Report the answer with no more significant figures than the accuracy of the given data.

26. General Procedure for Performing Calculations • Study the answer with technical judgment and common sense to determine whether or not it seems reasonable. • Once the solution has been completed, review that problem. Try to think of other ways of obtaining the same solution.

27. Logical Steps (1) State Given Data (2) State Results Desired (3) Draw free body diagrams (4) Perform Calculations / Analysis (5) Check Your Answers and Calculations • units are correct and consistent • equations are dimensionally homogeneous • no careless mistakes • numerical magnitudes are reasonable