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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.
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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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