B8 (a) A simply supported beam carries a point load Pat the mid span as shown in the following figure. y-axis L/2 y P Px y= (-4x + 3L) 48E1 L/2 x-axis The deflection y, along the beam is governed by the following equation: where Px y = (-4x + 3) for 0 x / 48E1 x is the distance from the support A; L is the total length of the beam; P is the point load acting at the mid span; E is the Young's modulus of the beam, and I is the moment of inertia of the section of beam. (Hint: L, P, E and I are constant parameters) (i) By using the application of differentiation, find the value of x where the deflection y is maximum; (5 marks) (ii) Find the maximum deflection, give your answer in terms of L, P, E and I. (5 marks) (b) Find the area bounded by the curve y = Px 48E1 (-4x + 3L) (10 marks) and the x-axis from x = 0 to x = L/2, give your answer in terms of L, P, E and I. B9 (a) Determine whether x+1 3x2-5x-2 is a proper or improper fraction. (12 marks) x+1 Hence, find the integrate : 3x-5x-2 dx. (b) A spring has a natural length of 30 mm. A 5 N force is required to compress the spring to a length of 25 mm. Calculate the work done in compressing the spring from 26 mm to 23 mm. (8 marks) 3