JUST THE MATHS SLIDES NUMBER. INTEGRATIN APPLICATINS (Second moments of n re (A)) b A.J.Hobson.. Introduction..2 The second moment of n re bout the -xis.. The second moment of n re bout the x-xis
UNIT. - INTEGRATIN APPLICATINS SECND MMENTS F AN AREA (A).. INTRDUCTIN Let R denote region (with re A) of the x-plne in crtesin co-ordintes. Let δa denote the re of smll element of this region. Then the second moment of R bout fixed line, l, not necessril in the plne of R, is given b lim δa R h2 δa, where h is the perpendiculr distnce from l of the element with re, δa.
l h δa R..2 THE SECND MMENT F AN AREA ABUT THE Y-AXIS Consider region in the first qudrnt of the x-plne, bounded b the x-xis, the lines x =, x = b nd the curve whose eqution is = f(x). δx x b 2
The region m be divided up into smll elements b using network consisting of neighbouring lines prllel to the -xis nd neighbouring lines prllel to the x-xis. All of the elements in nrrow strip, of width δx nd height (prllel to the -xis), hve the sme perpendiculr distnce, x, from the -xis. Hence, the second moment of this strip bout the -xis is x 2 (δx). The totl second moment of the region bout the -xis is given b lim δx x=b x= x2 δx = b x2 dx. Note: For region of the first qudrnt, bounded b the -xis, the lines = c, = d nd the curve whose eqution is x = g(), we m reverse the roles of x nd so tht the second moment bout the x-xis is given b d c 2 x d.
d δ c x EXAMPLES. Determine the second moment of rectngulr region with sides of lengths, nd b, bout the side of length b. Solution b x The second moment bout the -xis is given b x 2 b dx = x b = b. 4
2. Determine the second moment bout the -xis of the semi-circulr region, bounded in the first nd fourth qudrnts, b the -xis nd the circle whose eqution is x 2 + 2 = 2. Solution x There will be equl contributions from the upper nd lower hlves of the region. Hence, the second moment bout the -xis is given b 2 x 2 2 x 2 dx = 2 π 2 2 sin 2 θ. cos θ. cos θdθ, if we substitute x = sin θ. 5
This simplifies to 2 4 π 2 sin 2 2θ 4 dθ = 4 2 π 2 cos 4θ 2 dθ = 4 4 θ sin 4θ 4 π 2 = π4 8... THE SECND MMENT F AN AREA ABUT THE X-AXIS In the first exmple of the previous section, formul ws estblished for the second moment of rectngulr region bout one of its sides. This result m now be used to determine the second moment bout the x-xis of region, enclosed in the first qudrnt, b the x-xis, the lines x =, x = b nd the curve whose eqution is = f(x). 6
δx x b If nrrow strip, of width δx nd height, is regrded, pproximtel, s rectngle, its second moment bout the x-xis is δx. Hence, the second moment of the whole region bout the x-xis is given b lim δx x=b x= δx EXAMPLES = b dx.. Determine the second moment bout the x-xis of the region, bounded in the first qudrnt, b the x-xis, the -xis, the line x = nd the line whose eqution is = x +. 7
Solution x Second moment = (x + ) dx = (x + x 2 + x + ) dx = x 4 4 + x + + x2 2 + x = 4 + + 2 + = 5 4. 2. Determine the second moment bout the x-xis of the region, bounded in the first qudrnt b the x-xis, the -xis, the line x = nd the curve whose eqution is = xe x. 8
Solution x Second moment = = x ex x e x dx x 2 e x dx = x ex x 2ex + 2x ex dx = Tht is, x ex x 2ex + 2xex 9 x ex x2ex + 2xex 9 2 2ex 27 e x dx. = 4e + 2 8.2 9
Note: The second moment of n re bout certin xis is closel relted to its moment of inerti bout tht xis. In fct, for thin plte with uniform densit, ρ, the moment of inerti is ρ times the second moment of re since multipliction b ρ, of elements of re, converts them into elements of mss.