Advertisements
Advertisements
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.
Advertisements
Solution
\[\text { Given }: \hspace{0.167em} M = \frac{WL}{2}x - \frac{W}{2} x^2 \]
\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - 2 \times \frac{Wx}{2}\]
\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - Wx\]
\[\text { For maximum or minimum values of M, we must have }\]
\[\frac{dM}{dx} = 0\]
\[ \Rightarrow \frac{WL}{2} - Wx = 0\]
\[ \Rightarrow \frac{WL}{2} = Wx\]
\[ \Rightarrow x = \frac{L}{2}\]
\[\text { Now,} \]
\[\frac{d^2 M}{d x^2} = - W < 0\]
\[\text { So,M is maximum at }x = \frac{L}{2} . \]
APPEARS IN
RELATED QUESTIONS
f(x) = 4x2 + 4 on R .
f(x)=2x3 +5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 \[-\] 1 on R .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
`f(x)=xsqrt(1-x), x<=1` .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The minimum value of x loge x is equal to ____________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
