Example 18 - A beam is supported at its ends by supports 12 m - Parabola - Beam problem

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  1. Chapter 11 Class 11 Conic Sections
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Example 18 A beam is supported at its ends by supports which are 12 metres apart. Since the load is concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm? Beam is always facing upwards with the axis vertical Since, the axis is positive y-axis, its equation is x2 = 4ay First we find coordinates of point B Given Width of beam = 12 m Hence, AB = 12 m So, BC = 2 = 12 2 = 6 m Also, there is a deflection of 3 cm from centre So, OC = BD = 3 cm OC = BD = 3 cm = 3 100 m Hence point B is B(6, ) Now, Since point B(6, 3 100 ) lies on the parabola Putting x = 6, y = 3 100 in equation x2 = 4ay (6)2 = 4a 3 100 36 = 3 25 3 25 a = 36 a = 36 25 3 a = 12 25 a = 300 m Now, we need to find how far from the centre is the deflection 1 cm Hence RQ = 1 cm, We need to find OP QP = 3 cm 1 cm = 2cm = 2 100 m Let OP = x So, coordinates of point Q is Q(x, ) Since point Q lies on parabola it will satisfy the equation of parabola Equation of parabola is x2 = 4ay Putting x = x & y = 2 100 m & a = 300 m x2 = 4 (300) 2 100 x2 = 1200 2 100 x2 = 24 x = 24 = 6 4 = 2 6 Thus, the required distance is 2

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