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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
Advertisements
Solution
\[\text { Given }: \hspace{0.167em} M = \frac{Wx}{3} - \frac{W x^3}{3 L^2}\]
\[ \Rightarrow \frac{dM}{dx} = \frac{W}{3} - 3 \times \frac{W x^2}{3 L^2}\]
\[ \Rightarrow \frac{dM}{dx} = \frac{W}{3} - \frac{W x^2}{L^2}\]
\[\text { For maximum or minimum values of M, we must have }\]
\[\frac{dM}{dx} = 0\]
\[ \Rightarrow \frac{W}{3} - \frac{W x^2}{L^2} = 0\]
\[ \Rightarrow \frac{W}{3} = \frac{W x^2}{L^2}\]
\[ \Rightarrow x = \frac{L}{\sqrt{3}}\]
\[\text { Now }, \]
\[\frac{d^2 M}{d x^2} = - \frac{2Wx}{L^2} < 0\]
\[\text { So, M is maximum at } x = \frac{L}{\sqrt{3}} .\]
APPEARS IN
RELATED QUESTIONS
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) = (x - 1) (x + 2)2.
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
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 sufficient conditions for a point x = c to be a point of local maximum.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
