Share

Books Shortlist

Solution for 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 X 3 X − W 3 X 3 L 2 . Find the Point at Which M is Maximum in Case. - CBSE (Science) 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 $M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}$ .

Find the point at which M is maximum in case.

Solution

$\text { Given }: \hspace{0.167em} M = \frac{Wx}{3} - \frac{W x^3}{3 L^2}$

$\Rightarrow \frac{dM}{dx} = \frac{W}{3} - 3 \times \frac{W x^2}{3 L^2}$

$\Rightarrow \frac{dM}{dx} = \frac{W}{3} - \frac{W x^2}{L^2}$

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

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

$\Rightarrow \frac{W}{3} - \frac{W x^2}{L^2} = 0$

$\Rightarrow \frac{W}{3} = \frac{W x^2}{L^2}$

$\Rightarrow x = \frac{L}{\sqrt{3}}$

$\text { Now },$

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

$\text { So,M is maximum at } x = \frac{L}{\sqrt{3}} .$

Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

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 X 3 X − W 3 X 3 L 2 . Find the Point at Which M is Maximum in Case. Concept: Graph of Maxima and Minima.
S