JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson


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1 JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson Introduction First moment of n rc bout the yxis First moment of n rc bout the xxis The centroid of n rc Exercises Answers to exercises
2 UNIT INTEGRATION APPLICATIONS 6 FIRST MOMENTS OF AN ARC INTRODUCTION Suppose tht C denotes n rc (with length s) in the xyplne of crtesin coordintes; nd suppose tht δs is the length of smll element of this rc. Then the first moment of C bout fixed line, l, in the plne of C is given by lim δs C hδs, where h is the perpendiculr distnce, from l, of the element with length δs. h l δs C FIRST MOMENT OF AN ARC ABOUT THE YAXIS Let us consider n rc of the curve, whose eqution is y = f(x), joining two points, P nd Q, t x = nd x = b, respectively. 1
3 y δy P δs Q O δx x b The rc my divided up into smll elements of typicl length, δs, by using neighbouring points long the rc, seprted by typicl distnces of δx (prllel to the xxis) nd δy (prllel to the yxis). The first moment of ech element bout the yxis is x times the length of the element; tht is xδs, implying tht the totl first moment of the rc bout the yxis is given by lim δs C xδs. But, from Pythgors Theorem, δs ( ) (δx) + (δy) δy = 1 + δx, δx so tht the first moment of the rc becomes lim δx x=b x= ( ) δy x 1 + δx = δx b ( ) x 1 + dx. dx Note: If the curve is given prmetriclly by x = x(t), y = y(t),
4 then, using the sme principles s in Unit 13.4, we my conclude tht the first moment of the rc bout the yxis is given by t ± t 1 ) ( dx x + ( ), ccording s dx is positive or negtive FIRST MOMENT OF AN ARC ABOUT THE XAXIS () For n rc whose eqution is y = f(x), contined between x = nd x = b, the first moment bout the xxis will be b ( ) y 1 + dx. dx Note: If the curve is given prmetriclly by x = x(t), y = y(t), then, using the sme principles s in Unit 13.4, the first moment of the rc bout the xxis is given by t ± t 1 ) ( dx y + ( ), ccording s dx is positive or negtive. 3
5 (b) For n rc whose eqution is x = g(y), contined between y = c nd y = d, we my reverse the roles of x nd y in section so tht the first moment of the rc bout the xxis is given by d c ( ) dx y 1 +. d y S δy δs c R O δx x Note: If the curve is given prmetriclly by x = x(t), y = y(t), then, using the sme principles s in Unit 13.4, we my conclude tht the first moment of the rc bout the xxis is given by t ± t 1 ) ( dx y + ( ), ccording s is positive or negtive nd where t = t 1 when y = c nd t = t when y = d. 4
6 EXAMPLES 1. Determine the first moments bout the xxis nd the yxis of the rc of the circle, with eqution x + y =, lying in the first qudrnt. Solution y O x Using implicit differentition, we hve x + y dx =, nd hence, dx = x y. The first moment of the rc bout the yxis is therefore given by x But x + y = nd y = x. Hence, 1 + x y dx = x x y + y dx. first moment = x ] [ x dx = ( x ) =. By symmetry, the first moment of the rc bout the xxis will lso be. 5
7 . Determine the first moments bout the xxis nd the yxis of the first qudrnt rc of the curve with prmetric equtions Solution x = cos 3 θ, y = sin 3 θ. y O x Firstly, we hve dx dθ = 3cos θ sin θ nd dθ = 3sin θ cos θ. Hence, the first moment bout the xxis is given by π which, on using cos θ + sin θ 1, becomes y 9 cos 4 θsin θ + 9 sin 4 θcos θ dθ, π sin 3 θ.3 cos θ sin θ dθ = 3 π sin 4 θ cos θ dθ [ sin = 3 5 ] π θ 5 =
8 Similrly, the first moment of the rc bout the yxis is given by π ) ( ) ( dx π x + dθ = cos 3 θ.(3 cos θ sin θ) dθ dθ dθ = 3 π cos 4 θ sin θ dθ = 3 [ ] π cos5 θ 5 = 3 5, though, gin, this second result could be deduced, by symmetry, from the first THE CENTROID OF AN ARC Hving clculted the first moments of n rc bout both the xxis nd the yxis it is possible to determine point, (x, y), in the xyplne with the property tht () The first moment bout the yxis is given by sx, where s is the totl length of the rc; nd (b) The first moment bout the xxis is given by sy, where s is the totl length of the rc. The point is clled the centroid or the geometric centre of the rc nd, for n rc of the curve with eqution y = f(x), between x = nd x = b, its coordintes re given by x = b 1 x + ( ) dx dx b 1 + ( ) nd y = dx dx b 1 y + ( ) dx dx b 1 + ( ). dx dx Notes: (i) The first moment of n rc bout n xis through its centroid will, by definition, be zero. In prticulr, if we tke the yxis to be prllel to the given xis, with x s the perpendiculr distnce from n element, δs, to the yxis, the first moment bout the given xis will be C (x x)δs = C xδs x C δs = sx sx =. (ii) The centroid effectively tries to concentrte the whole rc t single point for the purposes of considering first moments. In prctice, it corresponds, for exmple, to the position of the centre of mss of thin wire with uniform density. 7
9 EXAMPLES 1. Determine the crtesin coordintes of the centroid of the rc of the circle, with eqution x + y =, lying in the first qudrnt. Solution y O x From n erlier exmple in this unit, we know tht the first moments of the rc bout the xxis nd the yxis re both equl to. Also, the length of the rc is π, which implies tht x = π nd y = π.. Determine the crtesin coordintes of the centroid of the first qudrnt rc of the curve with prmetric equtions x = cos 3 θ, y = sin 3 θ. 8
10 Solution y O x From n erlier exmple in this unit, we know tht dx dθ = 3cos θ sin θ nd dθ = 3sin θ cos θ nd tht the first moments of the rc bout the xxis nd the yxis re both equl to 3 5. Also, the length of the rc is given by π This simplifies to ) ( dx + dθ ( ) dθ = dθ π 9 cos 4 θsin θ + 9 sin 4 θcos θ dθ. π 3 cos θ sin θ dθ = 3 [ sin θ ] π = 3. Thus, x = 5 nd y = 5. 9
11 EXERCISES 1. Determine the first moment bout the yxis of the rc of the curve with eqution lying between x = nd x = 1. y = x,. Determine the first moment bout the xxis of the rc of the curve with eqution lying between y =.1 nd y =.5. x = 5y, 3. Determine the first moment bout the xxis of the rc of the curve with eqution lying between x = 3 nd x = 4. y = x, 4. Verify, using integrtion, tht the centroid of the stright line segment, defined by the eqution y = 3x +, from x = to x = 1, lies t its centre point. 5. Determine the crtesin coordintes of the centroid of the rc of the circle given prmetriclly by from θ = π 6 to θ = π For the curve whose eqution is x = 5 cos θ, y = 5 sin θ, 9y = x(3 x), show tht dx = 1 x x. Hence ( show tht the centroid of the first qudrnt rch of this curve lies t the point 7, )
12 ANSWERS TO EXERCISES x = 1 nd y = x = 15 π 4.77, y =. 11
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