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 = ﷐𝐴𝐵﷮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, ﷐𝟑﷮𝟏𝟎𝟎﷯) 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 = ﷐3𝑎 ﷮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, ﷐𝟐﷮𝟏𝟎𝟎﷯) 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﷯ 𝑚 Thus, the required distance is 2﷐﷮𝟔﷯ 𝒎

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