UNIVERSITY OF BOLTON OCD15 WESTERN INTERNATIONAL COLLEGE FZE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 016/017 ENGINEERING PRINCIPLES MODULE NO: AME4053 Date: Wednesday 4 May 017 Time: :00 4:00 INSTRUCTIONS TO CANDIDATES: There are SIX questions. Answer Two Questions from Part A and Two Questions from Part B. All questions carry equal marks. Marks for parts of questions are shown in brackets. Electronic calculators may be used provided the data and program storage memory is cleaned prior to the examination. CANDIDATES REQUIRE: Formula Sheet (attached)
Question 1 Page of 11 PART A a) The pressure P of the atmosphere at a height h above the ground level is given by P = P o e h C where P o is the pressure at the ground level and c is a constant. Determine the following when P o = 1.013 x 10 5 Pascal and c= 6.05 x 10 4. i. the initial rate of change of pressure (4 marks) ii. rate of change of pressure at a height of 1450 meters b) The fuel economy E of a car in miles per gallon, is given by the following equation: E = 1 +.10 10 v 3.80 10 6 v 4 where v is the speed of the car in miles per hour. Evaluate, correct to three significant figures, the most economical fuel consumption, and the speed at which it is achieved. (10 marks) c) Differentiate the following given equations: i. y = ii. x = 3 ln 5t cos t (t 3t+5) (3 marks) (3 marks) Total 5 marks Please turn the page Question
Page 3 of 11 a) The average value of complex voltage waveform is given by the following equation: V av = 1 π π (10 sin ωt 0 Evaluate V av correct to decimal places. + 3 sin 3ωt + sin 5ωt )d(ωt) b) A tank has a shape of an inverted circular cone of height 10m and base radius 4m. It is filled with water to a height of 8 m. Sketch the arrangement and find the work required to empty the tank by pumping all of the water to the top of tank. (10 marks) c) A stopper is attached to the end of a spring. When a force of Newton is exerted Question 3 on the stopper the spring compresses from 1 meter (its natural length) to 0.8 meter. Under same conditions what will be the net work required to stretch it from 1.1 meter to 1.5 meters? (10 marks) Total 5 marks a) For a beam, having a point load W and a span length l, the bending moment M of a beam at any section at a distance of x is given by dm = W(l x) Determine M in terms of x, given that at x = 0; M = 1 Wl (10 marks) Question 3 continued next page Question 3 continued..
Page 4 of 11 b) A differential equation relating the difference in tension T, pulley contact angle and coefficient of friction is given by dt dθ = μt When = 0, T = 150 N, and = 0.30 as slipping starts. i. Deduce a general solution for the above first order differential equation. ii. Determine the tension at the point of slipping when = radians. iii. Determine the value of when T is 300 N. PART B Total 5 marks Question 4 An inverted T-section is shown in Figure Question 4, determine the following: i. the centroid of the section (10 marks) ii. iii. the moment of inertia of the section about the xx axis through the centroid. (6 marks) the moment of inertia of the section about the yy axis through the centroid. (6 marks) iv. the radius of gyration (3 marks) Question 4 continued next page
Question 4 continued.. Page 5 of 11 All dimensions are given in mm. Figure Question 4.Inverted T- section Total 5 marks Question 5 a) A spring loaded with kg weight is extended 400mm when in equilibrium. The mass is pulled vertically downward through a further distance of 00mm and is then released from rest so that it oscillates about the equilibrium position. Determine : i. the stiffness constant k of the spring and time of oscillation in seconds ii. the velocity and acceleration when the weight is at a distance of 80mm below its equilibrium position. Question 5 continued next page
Question 5 continued.. Page 6 of 11 b) A shaft transmits 300kW power at 10 r.p.m.the allowable shear stress of the material is 70N/mm.The density of the material is 77kN/m 3. Determine: i. the necessary diameter of solid circular shaft (4 marks) ii. iii. Question 6 the necessary diameter of hollow circular section,the inside diameter being /3 of the external diameter. (6 marks) Also calculate the % saving in the material if hollow shaft is used. Total 5 marks A pull of 0N, inclined at 5 o to the horizontal plane, is required just to move a body on a rough horizontal plane. But the push required to move the body is 5N.If the push is inclined at 5 o to the horizontal, find the following: i. Draw the force body diagram for the systems shown in Figure Question 6a and Figure Question 6b. ( 5 marks) ii. the weight of the body (10 marks) iii. the coefficient of friction (10 marks) END OF QUESTIONS Total 5 marks
Vectors Page 7 of 11 FORMULA SHEET A.B = A Bcos Determinants x D x - y D y z D z -1 D Matrices 1 adja A D 1 X A B Series Un = a + (n 1) d Sn = n [a + (n 1) d] Un = ar n-1 Sn = n a(1 r ) 1 r S a 1 r Un = a + (n 1)d + 1 (n 1)(n )C Binomial (1 + x) n n(n 1) = 1 + nx + x......!
Validity x< 1 Partial Fractions Page 8 of 11 F(x) (x a)(x b) A B (x a) (x b) F(x) (x a)(x b) A B C (x a) (x b) (x b) Trigonometry sin x = sin x cos x cos x = cos x sin x cos x = cos x 1 cos x = 1 - sin x tan x = tan x 1 tan x sin x + cos x = 1 tan x + 1 = sec x cosec x = 1 + cot x Differentiation y = uv dy dv du = u + v (Product Rule) y = v u dy dy dt x dt dy du dv v u (Quotient Rule) v (ChainRule)
Integration dv du u uv v (By parts) Page 9 of 11 1 f (x) f(x) ln f(x) c Differential equations Linear differential equation dy/ + Py = Q Integrating factor is e P Solution is y IF = Q IF Centroid and nd Moments of Area bd Rectangle X = (b/), Y= (d/), A=bd 3 db 3 IXX = IYY = 1 1 Circle 4 πr I XX= 4 4 D Polar Jsolid = 3 Jhollow = π(d 4 d 4 )/3 For composite sections X = ΣAiXi ΣAi Y = ΣAiYi ΣAi Parallel Axis Theorem Ixx = IGG + Ah
IXX= (IXX)i +ΣAi(Yi Y) IYY= (IYY)i +ΣAi(Xi X) Energy and Momentum Potential Energy = mgh Page 10 of 11 Kinetic Energy Linear = ½ mv Angular=½ I Momentum Vibrations Linear Stiffness Linear= mv Angular= I k F Circular frequency n Frequency f n n 1 T n k m x r cost v r a x 1 f T T x r sin t M/I = σ/y =E/R
P= πnt T/J =Gθ/L =τ/r F=μ N Page 11 of 11