U اــــــــَــــــــيـــــــػ ٜـــــــــّدص بس وـــــــــــق ٗ N A J R A N I V E R S I T Y L n A e c t u r e o t e s n d E o x e r c i s e s n By Dr Abdulwahab Amrani GE 101 ٌؼٕ 101 Original text book: Engineering Mechanics -Statics, Twelfth Edition, R. C. Hibbeler, 2009. Lecture Notes and Exercises on STATICS D r Abdulwahab Amrani C O o u r s e b j e c t i v e s To understand and use the general ideas of force vectors and equilibrium of particle and rigid body. To understand and use the general ideas of structural analysis and internal force and friction. To understand and use the general ideas of center of gravity, centroids and moments of inertia. سشقَىا فاذٕأ :ه٘ؼ حٍاؼىا ٌٕٞافَىات ةىاؽىا ذٝٗضذ .ًاغظلأا ُاضذا خلاداؼٍ وؼ ٗ ٙ٘قىا وٞيؽذ حٞف ٞم  .كانرؼلاا ٗ حٞيخاذىا ٙ٘قىا ٗ ٜئاشّلإا وٞيؽ رىا  ًضػ ٗ وقصىا ضمشٍ ٗ ٜعذْٖىا ضمشَىا باغؼ حٞف ٞم  .س٘صقىا 2 Lecture Notes and Exercises on STATICS D r Abdulwahab Amrani C o n t e n t s Chapter page General principals 5 Force vectors 12 Equilibrium of a particle 36 Force system resultants 45 Equilibrium of a Rigid Body 61 Structural Analysis 78 Internal Forces 92 Friction 102 Center of Gravity and Centroid 109 Moments of Inertia 116 3 Lecture Notes and Exercises on STATICS D r Abdulwahab Amrani One 4 Lecture Notes and Exercises on STATICS D r Abdulwahab General Principals Amrani 1.1 Introduction The subject of statics developed very early in history because it’s principles can be formulated simply from measurements of geometry and force. Statics is the study of bodies that are at rest or move with constant velocity. We can consider statics as a special case of dynamics, in which the acceleration is zero. 1.2 Fundamental Concepts Before we begin our study, it is important to understand the meaning of certain fundamental concepts and principles. Length : Length is used to locate the position of a point in space and thereby describe the size of a physical system. Time: Although the principles of statics are time independent. This quantity plays an important role in the study of dynamics. Mass: Mass is a measure of a quantity of matter. Force: 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. A force is completely characterized by its magnitude, direction, and point of application. Particle Particle has a mass, but it size can be neglected. Rigid Body A rigid body can be considered as a combination of a large number of Particles Newton’s 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 (Fig.1-1). N 𝐹 = 0 i i=1 5 Lecture Notes and Exercises on STATICS D r Abdulwahab Amrani 𝐅 Fig. 1-1 a m Newton’s 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 ( Fig. 1-2). If “F” is applied to a particle or mass “m”, this law may be expressed mathematically as F = m . a Fig. 1-2 Newton’s third Law: The mutual forces of action between two particles are equal, opposite, and collinear (Fig. 1-3). Fig. 1-3 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. 𝑚 𝑚 1 2 𝐹 = 𝐺 𝑟2 Where F: Force of gravitational between the two particles. G: Universal constant of gravitation, according to experimental evidence. 6 Lecture Notes and Exercises on STATICS D r Abdulwahab m3 Amrani 𝐺 = 66.73 10−12 kg s2 m1, m2: mass of each of the two particles. r: distance between the two particles. Weight : Weight refers to the gravitational attraction of the earth on a body or quantity of mass. The weight of a particle having a mass is stated mathematically. 𝑊 = 𝑚𝑔 𝒎 Measurements give 𝒈 = 𝟗.𝟖𝟎𝟔𝟔 𝒔𝟐 Therefore, a body of mass 1 kg has a weight of 9.81 N, a 2 kg body weights 19.62 N, and so on (Fig. 1-4). Fig. 1-4 Units of Measurement :  SI units: The international System of units. Abbreviated SI is a modern version which has received worldwide recognition. As shown in Tab 1.1. The SI system defines length in meters (m), time in seconds (s), and mass in kilograms (kg). In the SI system the unit of force, the Newton is a derived unit. Thus, 1 Newton (N) is equal to a 𝑚 force required to give 1 kilogram of mass and acceleration of 1 . 𝑠2 .  US customary: In the U.S. Customary system of units (FPS) length is measured in feet (ft), time in seconds (s), and force in pounds (lb). The unit of mass, called a slug, 1 slug is equal to the 𝒇𝒕 amount of matter accelerated at 1 when acted upon by a force 𝒔𝟐 𝑙𝑏 𝑠2 of 1 lb (1 𝑠𝑙𝑢𝑔 = 1 ). 𝑓𝑡 7 Lecture Notes and Exercises on STATICS D r Abdulwahab Amran i Table 1.1 Systems of Units Name Length Time Mass Force International Systems of Units meter seconds kilogram Newton* 𝑘𝑔.𝑚 SI m s kg N 𝑠2 US Customary foot second Slug* pound 𝑙𝑏.𝑠2 FPS ft s lb ft *Derived unit Conversion of Units: Table 1.2 provides a set of direct conversion factors between FPS and SI units for the basic quantities. Also in the FPS system, recall that: 𝟏 𝐟𝐭=𝟏𝟐 𝐢𝐧 𝐢𝐧𝐜𝐡𝐞𝐬 𝟏 𝐦𝐢𝐥𝐞=𝟓𝟐𝟖𝟎 𝐟𝐭 𝟏 𝐤𝐩 𝐤𝐢𝐥𝐨 𝐩𝐨𝐮𝐧𝐝 =𝟏𝟎𝟎𝟎 𝐥𝐛 𝟏 𝐭𝐨𝐧=𝟐𝟎𝟎𝟎 𝐥𝐛 Table 1.2 Conversion factors Quantities Unit of Measurement (FPS) equals Unit of Measurement (SI) Force lb 4.448 N Mass slug 14.59 kg Length ft 0.3048 m Prefixes: 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. Some of the prefixes used in the SI system are shown in Table 1.3. Each represents a multiple or submultiples of a unit which, if (multiple) applied successively, moves the decimal point of a numerical quantity (submultiple ) to every third place. For example, 4000000N=4000kN (kilo- newton)=4MN (mega-newton), or 0.005m=5mm (milli·meter). Table 1.3 Prefixes Exponential Form Prefix SI Symbol Multiple 1 000 000 000 109 giga G 1 000 000 106 mega M 1 000 103 kilo K Submultiple 0.001 10-3 milli m 0.000 001 10-6 micro µ 0.000 000 001 10-9 nano n 8 Lecture Notes and Exercises on STATICS D r Abdulwahab Exercise 1.1: Amrani km m ft Convert 2 to . How many is this? h s s 𝑘𝑚 𝑚 𝑓𝑡 2 = 0.556 = 1.82 Ans: 𝑕 𝑠 𝑠 Exercise 1.2: slug Convert the quantities 300 lb.s and 52 to appropriate SI units. ft3 𝑠𝑙𝑢𝑔 𝑀𝑔 300 𝑙𝑏.𝑠 = 1.33 𝑘𝑁.𝑠 52 = 26.8 Ans: 𝑓𝑡" 𝑚3 Exercise 1.3: Evaluate each of the following and express with SI units having an appropriate prefix: (a) 50 mN 6 GN (b) 400 mm 0.6 MN 2 (c) 45 MN3 900 Gg Ans: 50 𝑚𝑁 6 𝐺𝑁 = 300 𝑘𝑁2 400 𝑚𝑚 0.6 𝑀𝑁 2 = 144 𝐺𝑚.𝑁2 45 𝑀𝑁3 = 50𝑘𝑁3 900𝐺𝑔 𝑘𝑔 Exercise 1.4: Round off the following numbers to three significant figures: (a) 4.65735 m (b) 55.578 s (c) 4555 N (d) 2768 kg 𝑎 4.66 𝑚 𝑏 55.6 𝑠 𝑐 4.56 𝑘𝑁 𝑑 = 2.77Mg Ans: Exercise 1.5: Represent each of the following combinations of units in the correct SI form using an appropriate prefix: (a) µMN (b) N/µm (c) MN/ks2 (d) kN/ms. 𝑀𝑁 𝑁 𝑀𝑁 : 𝑎 𝑁 𝑏 𝑐 𝑑 Ans 𝑚 𝑠2 𝑠 Exercise 1.6: Represent each of the following combinations of units in the correct SI form: (a) Mg/ms (b) N/mm (c) mN/(kg. µs). Mg Gg N kN mN kN a = b = c = Ans: ms s mm m kg.μs kg.s Exercise 1.7: A rocket has a mass of 250 103 slugs on earth. Specify (a) its mass in SI units and (b) its weight in SI units. If the rocket is on the moon, where the acceleration due to gravity is g =5.30 ft/s2, determine m to 3 significant figures (c) its weight in units, and (d) its mass in SI units. a 3.65 Gg b W = 35.8 MN c W = 5.89 MN m = m = 3.65 Gg Ans: e m m e Exercise 1.8: If a car is traveling at 55 mi/h, determine its speed in kilometers per hour and meters per second. 𝑘𝑚 𝑚 𝑎 88.514 𝑏 24.6 Ans: 𝑕 𝑠 Exercise 1.9: The Pascal (Pa) is actually a very small units of pressure. To show this, convert 1 Pa=1 N/m2 to lb/ft2. Atmospheric pressure at sea level is 14.7 lb/in2. How many Pascals is this? 𝑎 1 𝑃𝑎 = 20.9 10−3 𝑙𝑏 𝑏 1 𝐴𝑇𝑀 = 101.34 𝑘𝑃𝑎 Ans: 𝑓𝑡2 9 Lecture Notes and Exercises on STATICS D r Abdulwahab Exercise 1.10: Amrani Two particles have a mass of 8 kg and 12 kg, respectively. If they are 800 mm apart, determine the force of gravity acting between them. Compare this result with the weight of each particle. 𝑎 𝐹 = 10.0 𝑛𝑁 Ans: Exercise 1.11: Determines the mass in kilograms of an object that has a weight of: (a) 20 mN (b) 150 kN (c) 60 MN Ans: 𝑎 𝑚 = 2.04 𝑔 𝑏 𝑚 = 15.3 𝑀𝑔 𝑐 𝑚 = 6.12 𝐺𝑔 10