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Chapter 1 : Introduction Department of Mechanical Engineering Introduction What is mechanics? Fundamental quantities of mechanics Units of Measurements Dimensional consideration Newton’s law Mass and Weight Significance of numerical results Free body diagram Department of Mechanical Engineering What is mechanics? The study of response of bodies to action of forces Mechanics: – Rigid body mechanics – Mechanics of deformable bodies – Fluid mechanics Key subjects: – Forces, reactions, and stresses – Strength of materials – Structure mechanics – Free Body Diagrams (FBD) Department of Mechanical Engineering Some Quantities of Mechanics Space and Length – Space = a geometric region where the physical event takes place coordinate systems – Length = a measure of the size of the space Time: – The interval between two events – Examples: before and after the application of a load Matter – Substance that occupies the space or a body – Contributes to the “resistance” of the body under loading – Mass = the quantification of matter A force – An action of one body upon another body – The two bodies may be in contact or separated – Gravitational force separated bodies – Action and reaction bodies in contacts A particle: – It is often used as a representation of a body – It may have mass but not size or shape Rigid-body concepts – Idealization of bodies/particles/collection of particles – The shape and size of a rigid body remain constant under loadings – In reality, bodies will be deformed and/or displaced under loadings Department of Mechanical Engineering Units of Measurements Measurement of physical quantities require “standards” – “Standard” is arbitrarily defined (but must be agreed) – Two systems of “standards” are commonly used The US Customary System of Units – USCS – British units – Foot – pound (for force) - second The International System of Units – SI units – MKS units – Meter-kilogram-second – Metric system Department of Mechanical Engineering International System of Units Base Units (Table 1-2) – Meter for length, kg for mass, etc. Supplementary Units (Table 1-3) – Only rad and sterad for angles – Often regarded as non dimensional Derived Units (Table 1-4) – Based on the physical law of the quantity – Combinations of base units and/or supplementary units – m2 for area, m/s for speed, etc Department of Mechanical Engineering Dimensional Homogeneity An equation is dimensionally homogeneous if the form of the equation does not depend on the units of measurement. – Example: These equation are valid regardless of the unit system that is being used (provided that the unit system is NOT mixed) » Area = width x length » s = a x t2 s = distance, a = acceleration, t = time » Force = mass x acceleration Department of Mechanical Engineering Dimensional consideration Physical quantities can be expressed dimensionally in terms of three fundamental quantities: mass (M), length (L), and time (T) The dimensions of quantities other than the fundamental quantities follow from the physical laws Examples: – Area L2 – Velocity L/T – Force ML/T2 – Work = Force x Distance ML2/T2 – Power = Rate of Work ML2/T3 Department of Mechanical Engineering Commonly used physical quantities and their dimensions Department of Mechanical Engineering Dimensional Homogeneity Also implies that the LHS and RHS of the equation must have the same dimension Example: – The formulation of the non-dimensional angle of twist (θ ) of a circular beam is given by » T = moment of force = ML2/T2 » L = length of the beam = L TL » J = Torsional inertia = L4 θ= 2 » G = Shearing modulus = M/L/T GJ ≈ θ = angle of twist = dimensionless – We can show that the LHS and RHS have the same dimension TL ML2 / T 2 L θ= = 4 =1 2 GJ L M /( LT ) Department of Mechanical Engineering Newton’s Laws of Motion Law 1: In the absence of external forces, a particle originally at rest or moving with constant velocity will remain at rest or continue to move with a constant velocity along a straight line ∑ v = constant Law 2: If an external force acts on a particle, the particle will be accelerated in the direction of the force and the magnitude of the acceleration will be directly proportional to the force and inversely proportional to the mass of particle 1 ∑ F = 0 ⇒ a = 0, F ∝a a∝ m Law 3: For every action there is an equal and opposite reaction. The forces of action and reaction between bodies are equal in the magnitude, opposite in direction, and collinear Action = −Reaction Department of Mechanical Engineering Law of Gravitation Law that governs the mutual attraction between two bodies (in contact or separated) F = Force magnitude G = universal gravitational constant r = distance between the center of the mass m = mass of the body m1m2 F =G 2 r G = 3.439e-8 ft3/(slug.s2) G = 6.673e-11 m3/(kg.s2) e+n = 10+n r = center of the mass m1 m2 Department of Mechanical Engineering Example: Determine the magnitude of gravitational force exerted between two bodies of 20 and 50 kgs when they are 1 m, 1 cm and 1 mm apart. – G = 6.673e-11 m3/(kg.s2) Answer: – F = G 20*50/12 = 6.673e-8 N – F = G 20*50/0.012 = G 20*50/1e-22 = 6.673e-4 N – F = G 20*50/0.0012 = G 20*50/1e-32 = 6.673e-2 N » N = Newton = kg.m/s2 Gravitational force in general can be ignored Department of Mechanical Engineering Mass and Weight Mass (m) – The absolute quantification of matter occupied in a body – It is independent of the position of the body in space and the surrounding forces Weight (W) – Gravitational attraction exerted by the earth on a body – Generally speaking, it depends on the location on the earth surface m1m2 F =G 2 r Set: m1 = mass of the earth (me) m2 = any mass on the earth surface (m) r = mean radius of the earth (re) g = G me/re2 W = mg g = 9.81 m/s2 g = 32.2 ft/s2 Department of Mechanical Engineering Example: The weight of an object on the earth surface was found to be 98100 N. Find its weight on the moon and on 200,000 km above the earth surface (half way to the moon). – Mass of the earth = me = 5.976e24 kg – Mean radius of the earth = re = 6370 km – Mass of the moon = mm = 7.350e22 kg – Mean radius of the moon = rm = 1740 km – G = 6.673e-11 m3/(kg.s2) m1m2 F =G 2 Answer: r – Utilize the gravitational law – Remember: the mass of the object (m) is constant everywhere Department of Mechanical Engineering Answer mass of the object is given by m = Wg = 98100 = 10000kg 9.81 The weight on 200000 km above the earth surface is given by m×m The F =W = G e 2 (re + s ) 1.00e4 × 5.976e24 (6.370e6 + 2.00e5) 2 = 92384.89 → 9.24e4 or 92400 N = 6.673e − 11× The weight of the object on the moon surface is F =W = G m × mm 2 rm 1.00e4 × 7.350e22 1.740e6 2 = 16199.81 → 1.620e4 or 16200 N = 6.673e − 11× Department of Mechanical Engineering Significance of Numerical results The accuracy of the known physical data – The accuracy of analysis results cannot be more than the input of the analysis The accuracy of the model – The accuracy of analysis results is limited by the choice of the model The accuracy of the computations – The number of significant figures given by the computers does not represent the true accuracy because of the two limitation above Bottom lines: – Limit the results to four significant figures – Try to avoid round-off error when performing calculation Department of Mechanical Engineering Significant figures Please read last paragraph on p.18 Retain four significant figures if the leading figure is small: 12.34 and 2.345 Retain three significant figures if the leading figure is large: 0.0876 Rounded down if the digits being dropped < half of the last significant figure retained: 7654 7650 because 4 < ½*10 Rounded up if the digits being dropped > half of the last significant figure retained: 123.456 123.5 because 0.056 > ½*0.1 If the digits being dropped = ½*last figure retained, then – Keep it unchanged when the last figure is an odd number – Rounded up if the last figure is an even number – Example: 12,345 and 12,355 both become 12,350 6780 and Department of Mechanical Engineering Reading assignments Section 1-5: Problem Solving Section 1-6: Significance of Numerical Results Department of Mechanical Engineering Free body diagram (FBD) A collection of bodies of interest separated from all other interacting bodies and with all external forces applied Pushing force Frictions Frictions Weight Department of Mechanical Engineering