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A Beam is Supported at the Two End and is Uniformly Loaded. the Bending Moment M at a Distance X from One End is Given by M = W L 2 X − W 2 X 2 . Find the Point at Which M is Maximum in Case. - CBSE (Arts) Class 12 - Mathematics

Question

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by $M = \frac{WL}{2}x - \frac{W}{2} x^2$ .

Find the point at which M is maximum in a given case.

Solution

$\text { Given }: \hspace{0.167em} M = \frac{WL}{2}x - \frac{W}{2} x^2$

$\Rightarrow \frac{dM}{dx} = \frac{WL}{2} - 2 \times \frac{Wx}{2}$

$\Rightarrow \frac{dM}{dx} = \frac{WL}{2} - Wx$

$\text { For maximum or minimum values of M, we must have }$

$\frac{dM}{dx} = 0$

$\Rightarrow \frac{WL}{2} - Wx = 0$

$\Rightarrow \frac{WL}{2} = Wx$

$\Rightarrow x = \frac{L}{2}$

$\text { Now,}$

$\frac{d^2 M}{d x^2} = - W < 0$

$\text { So,M is maximum at }x = \frac{L}{2} .$

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Solution A Beam is Supported at the Two End and is Uniformly Loaded. the Bending Moment M at a Distance X from One End is Given by M = W L 2 X − W 2 X 2 . Find the Point at Which M is Maximum in Case. Concept: Graph of Maxima and Minima.
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