Statics, UniKL Malaysian Institute of Aviation Technology PDF
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Uploaded by HealthfulRutherfordium
UniKL Malaysian Institute of Aviation Technology
2016
Wan Nur Shaqella Bte Wan Abdul Razak
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Summary
This document is a set of lecture notes on statics. It covers topics like forces, moments and couples, vector representation, center of gravity, elements of theories, and pressure/buoyancy in liquids, and is suitable for an undergraduate level course in engineering.
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2. STATICS February 11, 2016 LEARNING OUTCOMES On compleAon of this topic you should be able to: Describe about staAcs. 1. Forces, moments and couples, representaAon as vectors. 2....
2. STATICS February 11, 2016 LEARNING OUTCOMES On compleAon of this topic you should be able to: Describe about staAcs. 1. Forces, moments and couples, representaAon as vectors. 2. Centre of gravity. 3. Elements of theories. 4. Nature of properAes. 5. Pressure and buoyancy in liquids (Barometer). Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 2 StaAc qIf a Force is applied to a body it will cause that body to move in the direc8on of the applied force, a force has both magnitude (size) and direcAon. qSome forces require contact between the two objects: -‐ e.g. the force of fric8on between car 8res and the road as the car corners. qSome forces do not require contact: -‐ e.g. the force between two magnets. qStaAcs is used to describe study of bodies at rest when forces are balanced. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 3 2.1 FORCES Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 4 Force q Force – anything that tends to cause mo8on, change of mo8on, stop mo8on or prevent mo8on. q Work is the product of a force applied to an object 8mes the distance the object moves. q Force has a unit of Newtons (N). q One Newton is defined as the force which gives a mass of 1 kg an accelera8on (or decelera8on) of 1 m/s2, i.e. 1 N = 1 kg m/s2. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 5 Forces q Normally more than one force acts on an object. q An object res8ng on a table is pulled down by its weight W and pushed back upwards by a force R due to the table suppor8ng it. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 6 2.1 SCALAR & VECTOR QUANTITY (Cont.) Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 7 SCALAR QUANTITY VECTOR QUANTITY v Quantity (by a single number) v Quantity (by a number / magnitude and a direction) v Number with units (+ve, -ve, 0) v Magnitude of vector: |F| = F always +ve v Example: length, time, v Example: Force, momentum, temperature, mass, density, velocity, displacement, volume acceleration v Acceptable symbol for vector is F Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 8 Vector AddiAon If a particles undergoes a displacement A, followed by a second displacement B. The final result is the same as if the particle had started at the same initial point and undergone a single displacement C. We call the displacement C as Vector Sum or Resultant. B A C Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 9 Two or more forces may act upon the same point so producing a resultant force. If the forces act in the same straight line the resultant is found by simple subtracAon or addiAon. If the forces are do not act in a straight line then they can be added together using the ‘parallelogram law’. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 10 Example A UniKL Miat student walks 12 km east one day and 5km east the following day. Find the resultant vector for the journey of the student? First day 12 km Second day 5 km 17 km to the east A UniKL Miat student walks 12km east one day and 5km west the next day. Find the resultant vector for the journey of the student? Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 11 EQUILIBRANT Force 1. A single force that can hold the original system of forces in equilibrium is known as the EQUILIBRANT. 2. It is equal in magnitude to the resultant but it is opposite in sense. B A A C B Equilibrant Force Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 12 Vectors in 2 Dimension form. (axis – x and axis – y) A vector in two dimensions may be resolved into two component vectors acting along any two mutually perpendicular directions. +y A = Ax + Ay Ax = Acos θ Ay = Asin θ Ay A Magnitude, |A| = √(Ax2 + Ay2) θ Direction, tan θ = Ay / Ax +x Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 13 Component vector along x and y axis depend on the angle, θ Bx – NegaAve Ax – PosiAve By -‐ PosiAve Ay -‐ PosiAve B A Cx – NegaAve Dx – PosiAve C D Cy -‐ NegaAve Dy -‐ NegaAve Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 14 Obtain the Resultant Force of the Following Vectors? |B| = 180 N θ = 25o |A| = 150 N θ = 20o Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 15 Obtain the Resultant Force of the Following Vectors? |A| = 60 N θ = 35o θ = 30o |B| = 80 N Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 16 Obtain the Resultant Force of the Following Vectors? |C| = 60 N |A| = 160 N θ = 40o θ = 35o θ = 30o |B| = 80 N Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 17 2.1 MOMENTS AND COUPLES (Cont.) Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 18 Moment of a Force Ø A force can also be used to produce rota8on, as occurs when opening a door or 8ghtening a nut with a spanner. Ø This turning effect of the force is known as “the moment of the force”. Ø It depends on the magnitude of the force and a distance called the lever arm. This is the perpendicular distance from the force to the axis of rota8on. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 19 Moment Moment (Nm) = Magnitude of the force (N) x Perpendicular distance (d) Applying the force in such a way that its line of acAon passes through the pivot will not produce a turning effect. In SI units, Newton metres = Newton x metres Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 20 Line of acAon of the Applied force force Pivot Pivot Line of acAon of the force Applied force Pivot If the force causes the lever to move in a clockwise direc8on, the moment is said to be a clockwise moment, and vice versa. If the force is inclined, the turning effect is reduced i.e. moment is reduced because the perpendicular distance is reduced. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 21 Example In the diagram above a force of 5 N is applied at a distance of 3 m from the fulcrum, therefore: Moment = 5 N x 3 m = 15 N m Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 22 Moments and Equilibrium v Equilibrium is where all the forces and all the moments ac8ng on a body cancel each other and the net effect on the body is zero. v In other words it will not move if it is in a state of rest, and if in mo8on it will not slow down or accelerate or change direc8on. P1 P2 S1 S2 Anticlockwise Clockwise tendency tendency Ø The product P2 x S2 produces a clockwise moment about the pivot and the product P1 x S1 produces an anA-‐clockwise moment about the pivot. Ø For equilibrium of rota8on, these two moments must be equal. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 23 Example of One Unknown Force 1. In this case the requirement is to balance the arrangement in figure 15 by determining the unknown force (P). Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 24 Example of Pivot LocaAon 2. A uniform bar AB (figure 16) 7m long has forces of; 25N at a point 0.5m from A, 12N at a point 3m from A, and 12N at a point 1m from B applied to it. Find the posi8on of the pivot which will allow the beam to balance, i.e. be in the state of equilibrium. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 25 Example of Mass & Forces Pivot LocaAon 3. A uniform beam AB, 4m long and 200N weight has forces of 125N and 20N applied respec8vely to its ends A and B. Find the point about which the beam will balance. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 26 Principle of Moments ‘When a body is in equilibrium under the ac5on of a number of forces, the sum of the clockwise moments about any point is equal to the sum of the an5clockwise moments about that point.’ a) Type 1 – Beam balances where arms are of equal length. b) Type 2 – Lever arrangement can best be seen in design of a wheelbarrow. c) Type 3 – Large effort moves through small distance to overcome small load, which moves through a large distance. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 27 Couple Ø A special case of moments is a couple. A couple consists of two parallel forces that are equal in magnitude, opposite in sense and do not share a line of ac8on. Ø It does not produce any translaAon, only rotaAon. The resultant force of a couple is zero. BUT, the resultant of a couple is not zero; it is a pure moment. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 28 Example q In some situa8ons, for example the winding up of a clockwork mechanism the forces that are applied to the winding key are equal in magnitude but opposite in sense. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 29 q In this case the resultant force on the pivot is zero and there is only pure rota8on present with no tendency for the pivot to move sideways. The value of the resultant moment ( P x d ) produces rota8on. q Such arrangement of forces is called a ‘COUPLE’ and the resultant moment of a couple is called a TORQUE. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 30 Example of Resultant Moment Ø Calculate the resultant moment of a pivot ac8ng on a bell crank lever, refer to diagram below. AO = 100 mm OC = 20 mm BC = 20 mm Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 31 2.2 CENTRE OF GRAVITY Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 32 Centre of Gravity Ø Gravity is a force which is always present and is a pulling force in the direc8on of the center of the earth. Ø The centre of gravity is the force acts on every body through an imaginary point . A point where all the weight of a body appears to be concentrated. (total weight can be considered to act through that datum posi8on ) Ø In flight, both airplanes and rockets rotate about their centre of gravity. Determining the centre of gravity is very important for any flying object. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 33 Example of Centroid Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 34 Stability / Balancing The lower the C of G, the stable an object is. The wider the base, the more stable an object is – C of G towards the base. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 35 Ø The point of O is the C of G of the rod Ø Only at the par8cular point O, the rod can stay in a horizontal posi8on. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 36 Ø When force applied to C of G, the body will not rotate. Ø But if the force is applied offset of the C of G, the body will rotate, or torque will produced. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 37 C of G of an aircrai Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 38 v Similar to aircrai, force applied will be acted through the C of G, resul8ng in torque. v The aircrai rotate about its C of G. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 39 The Importance of C of G q To ensure the aircrai is safe to fly, the center-‐of-‐gravity must fall within specified limits established by the manufacturer. q To ensure the C of G range – C of G limits are specified longitudinal (forward and ai) and/or lateral (lei and right) limits within which the aircrai's center of gravity must be located during flight. q To evenly load the aircrai – equipments, passengers, baggage, cargo, fuel, etc. q So that C of G range will not be exceeded – prevent aircrai unstable during flight. q Also affects C of G in flight – fuel usage, passengers’ movement, etc. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 40 2.3 ELEMENTS OF THEORIES Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 41 Stress Ø If force is exerted on a body, there will be mechanical pressure ac8ng on the body which is called the stress. Ø A body with having twice the size of other body subjected to a force, it will be stronger and less likely to fail due to applied the applied force. Ø So, stress is said : Stress = or σ = *units : Newton metre -‐2 , Nm-‐2 Ø Components will fail due to over-‐stressed, not over-‐loaded. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 42 Example Eg. A tennis ball sealed from atmospheric pressure. So, as long as the external forces ac8ng on it does not exceed the internal forces, the ball will maintain its shape. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 43 Forces applied to the body will cause distor8on of the body and change to the material’s cross-‐sec8onal area ; eg. Tensile Forces will cause elonga8on . Compressive Force will cause reduc8on in dimension. Most material have elas8c proper8es ( it will to return to its original shape aier the force is removed ) -‐ provided forces does not exceed limit of elas8city. There are 5 types of stress in mechanical bodies : i. Tension ii. Compression iii. Torsion iv. Bending v. Shear Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 44 Tensile v The force that tends to pull an object apart v Flexible steel cable used in aircrai control systems is an example of a component that is in designed to withstand tension loads. Compression v The resistance to an external force that tries to push an object together. v Example: aircrai rivets. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 45 Torsion v Torsional stress is applied to a material when it is twisted. v Torsion is actually a combina8on of both tension and compression v Example: an engine crankshai. Bending v In flight, the force of lii tries to bend an aircrai's wing upward. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 46 Shear v Combina8on of tension and compression is the shear stress, which tries to slide an object apart. v Shear stress exists in a clevis bolt when it is used to connect a cable to a sta8onary part of a structure. v A fork filng, such as drawn below, is fastened onto one end of the cable, and an eye is fastened to the structure. The fork and eye are held together by a clevis bolt. v When the cable is pulled there is a shearing ac8on that tries to slide the bolt apart. This is a special form of tensile stress inside the bolt caused by the fork pulling in one direc8on and the eye pulling in the other. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 47 Strain u Stress is a force inside the object caused by an external force. u If the outside force is great enough to cause the object to change its shape or size, the object is not only under stress, but is also strained. u If a length of elas8c is pulled, it stretches. If the pull is increases, it stretches more; if the pull is reduced, it contracts. Hooke’s law states that the amount of stretch (elongation) is proportional to the applied force. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 48 v The graph below shows how stress varies with stress when a steel wire is stretched un8l it breaks. v Strain can be defined as the degree of distor8on then has to be the actual distor8on divided by the original length (in other words, elonga8on per unit length). Strain, ε = change in dimension / original dimension (No units). Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 49 Example Tensile strain If a cable of 10 m length is loaded with a 100 kg weight so that it is stretched to 11 m, what is the strain placed on the cable? Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 50 Example Compressive strain A 25 cm rod is subjected to a compressive load so that its length changes by 5 mm. How much strain is the rod under when loaded? Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 51 Shear strain Torsion strain q When the applied load causes q Form of shear stress resul8ng one 'layer' of material to from a twis8ng ac8on. move rela8ve to the adjacent layers. q If a torque, or twis8ng ac8on is applied to the bar shown, one end will twist, or deflect rela8ve to the other end. q Twist will be propor8onal to the applied torque. 52 Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 2.4 NATURE OF PROPERTIES Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 53 ProperAes of Solid v Strength – A strong material requires a strong force to break it. The strength of some materials depends on how the force is applied. – For example, concrete is strong when compressed but weak when stretched, i.e. in tension. v SAffness – A s8ff material resists forces which try to change its shape or size. It is not flexible. v ElasAcity – When the force distor8ng a substance is removed, and that substance has a strong tendency to return to its original shape, it is said to be elas8c. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 54 ProperAes of Solid v Toughness – This is the ability of a substance to resist breakage when deforming or impact forces are applied to it. Hard substances are usually tough, many soier substances are tough e.g. hammer heads. v Hardness – A hard substance has a high resistance to indenta8on, or to any ac8on tending to penetrate its surface. In other words, hardness is the ability of a material to resist scratching, indenta8on or penetra8on. The harder a material the more difficult it is to scratch it, dent it or cut it. v Brilleness – Briqle substances break with liqle or no change of shape. In most applica8ons, especially where sudden impact-‐type forces are applied, briqleness is undesirable. At room temperature and below, glass, cast iron, and very hard steel are example of briqle materials. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 55 ProperAes of Solid v Malleability – Malleable materials can be beaten, rolled, or pressed into shape without fracture e.g. red hot steel. Malleable metal can be shaped into a design by hilng it. It could also lose that design easily by being hit against countertops, cash register drawers, and other hard surfaces. v DucAlity – Duc8le materials can be stretched into new shapes without pulling them apart, and keep their new shape aier stretching force is removed. v PlasAcity – Plas8city is the ability of a material to have its shape permanently changed without fracturing by stretching, squashing or twis8ng. In other words, plas8city is described as a material that does not spring back to its original shape when the load is removed. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 56 ProperAes of Fluid v Viscosity – As the molecules of a liquid move about due to thermal energy, the aqrac8ve forces between them try to slow the mo8on down. The stronger the forces are the more impediments there is to flow. Such resistance to the flow of a liquid is called viscosity. Viscosity is defined, as the amount of force one layer of liquid of unit area will exert on an adjacent layer. v Surface Tension – The molecules of a liquid within the body of the liquid are subjected to forces from all direc8ons. The molecules at the surface are subjected to aqrac8ve forces from within and to the sides. – However there are no forces from the outer side of the surface to balance the others. This places the surface molecules under a kind of tension. This surface tension tends to cause the surface molecules to move together and make the surface area as small as possible. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 57 Surface Tension This suggests that the surface of a liquid behaves as if it is covered with an elas8c skin that is trying to shrink. The surface tension can be reduced if the liquid is ‘contaminated’, adding a detergent to the water will cause our needle to sink. In a liquid, the molecules sAll parAally bond together and prevents liquid from spreading nag expanding out. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 58 Example of Surface Tension Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 59 ProperAes of Fluid v EvaporaAon – A change of a substance from one state of maqer to another is called a phase change. The phase change from liquid to a gas is called in general vaporiza8on. – Vaporiza8on from the surface of a liquid at any temperature is called evapora8on. v Boiling Point – The molecules are in mo8on with a range of energies. Some can escape when they reach the surface if they have enough energy. Any liquid will have within its body however, microscopic bubbles. These may be due to dissolved air or momentary pockets of vapour. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 60 2.5 PRESSURE AND BUOYANCY IN LIQUIDS (BAROMETERS) Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 61 Pressure Ø The equivalent term associated with fluids is pressure: Pressure (P) = Force (F)/ Area (A) Ø Pressure is the internal reac8on or resistance to that external force. Ø SI system for pressure is 1 Pa = 1N/m2 Pascal’s Law : “Pressure acts equally and in all directions throughout that fluid.” Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 62 Atmospheric Pressure The atmosphere is the whole mass of air surrounding the earth. The surface of the earth is at the boqom of an atmospheric sea. The standard atmospheric pressure is measured in various units: 𝟏 𝒂𝒕𝒎𝒐𝒔𝒑𝒉𝒆𝒓𝒆 = 𝟕𝟔𝟎𝒎𝒎𝑯𝒈 = 𝟐𝟗. 𝟗𝟐𝒊𝒏𝑯𝒈 = 𝟏𝟒. 𝟕𝒍𝒃/𝒊𝒏𝟐 = 𝟏𝟎𝟏. 𝟑𝒌𝑷𝒂 Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 63 Measurement of Atmospheric Pressure Atmospheric pressure is typically measured in inches of mercury (in.Hg.) by a mercurial barometer. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 64 Barometer Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 65 Gauge Pressure Ø Gauge Pressure is the reading taken directly from the gauge devices Ø It is a pressure rela8ve to the ambient pressure. Ø Gauge pressure is used to measure engine oil pressure, hydraulic pressure and other operaAonal pressures built up by pumps. Ø This is because atmospheric pressure acts on the fluid as it enters and as it leaves the pump – only the pressure above atmospheric is of interest. Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 66 PRESSURE GAUGE Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 67 Absolute Pressure Absolute Pressure is the sum of the available atmospheric pressure and the gauge pressure. Absolute Pressure (PSIA) = Gauge Pressure + Atmospheric Pressure Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 68 CalculaAon Example : Given (Gauge Pressure) = 150 psig (Atmospheric Pressure) = 14.7psi Absolute Pressure = 150 psig + 14.7 psi = 164.7 PSIA Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 69 Buoyancy Archimedes’ Principle states that when an object is submerged in a liquid, the object displaces a volume of liquid equal to its volume and is supported by a force equal to the weight of the liquid displaced. THE BUOYANCY OF A SUBMERGED BODY = WEIGHT OF DISPLACED LIQUID – WEIGHT OF THE BODY 1. The body will float-‐-‐if the buoyancy is posiAve 2.The body will sink-‐-‐if the buoyancy is negaAve 3.The body will be stuck-‐-‐if the buoyancy is neutral Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 70 Buoyancy Archimedes Principle q When an object is submerged in a liquid, the object displaced a volume of liquid equal to its volume and is supported by a force equal to the weight of the liquid it displaced. q The buoyant force of an object which is submerged in a fluid is equal to the weight of the fluid displaced by the object. q A net upward ver8cal force results because pressure increases with depth and the pressure forces ac8ng from below are larger than the pressure forces ac8ng from above. Buoyant Force, FB = ρgV Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 71 Archimedes Principle “Any object completely or parAally submerged in a fluid experiences an upward force equal in magnitude to the weight of the fluid displaced by the object.” Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 72 THIS EXPLAINS WHY BIG NAVAL SHIP CAN FLOAT !!!!!! A steel ship can encompass a great deal of empty space and so have a large volume and a relatively small density. Weight of ship = weight of water displaced Prepared By: Wan Nur Shaqella Bte Wan Abdul Razak 73
