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(a) What is the area of a thin slice ofSat positionxwith width dx?(b) What is the mass of a small piece ofRat positionxwith length dx?(c) What is the total area ofS?(d) What is the total mass ofR?(e) What is thex-coordinate of the centroid ofS?(f) What is the centre of mass ofR?In Questions8through10, you will derive the formulas for the centre of mass of a rod of variable density, andthe centroid of a two-dimensional region using vertical slices (Equations 2.3.2 and 2.3.3 in the CLP–II text).Knowing the equations by heart will allow you to answer many questions in this section; understandingwhere they came from will you allow to generalize their ideas to answer even more questions.75
APPLICATIONS OFINTEGRATION2.3 CENTRE OFMASS ANDTORQUEQ: SupposeRis a straight, thin rod with densityρ(x)at a positionx. Let the leftendpoint ofRlie atx=a, and the right endpoint lie atx=b.(a) To approximate the centre of mass ofR, imagine chopping it intonpieces of equallength, and approximating the mass of each piece using the density at its midpoint.Give your approximation for the centre of mass in sigma notation.m1m2mnab(b) Take the limit asngoes to infinity of your approximation in part (a), and express theresult using a definite integral.Q: SupposeSis a two-dimensional object and at (horizontal) positionxits height isT(x)´B(x). Its leftmost point is at positionx=a, and its rightmost point is at positionx=b.To approximate thex-coordinate of the centroid ofS, we imagine it as a straight, thin rodR, where the mass ofRfromaďxďbis equal to the area ofSfromaďxďb.(a) IfSis the sheet shown below, sketchRas a rod with the same horizontal length,shaded darker whenRis denser, and lighter whenRis less dense.xyT(x)B(x)ab(b) If we cutSinto strips of very small width dx, what is the area of the strip at positionx?(c) Using your answer from (b), what is the densityρ(x)ofRat positionx?(d) Using your result from Question8(b), give thex-coordinate of the centroid ofS. Youranswer will be in terms ofa,b,T(X), andB(x).Q: SupposeSis flat sheet with uniform density, and at (horizontal) positionxitsheight isT(x)´B(x). Its leftmost point is at positionx=a, and its rightmost point is atpositionx=b.To approximate they-coordinate of the centroid ofS, we imagine it as a straight, thin,vertical rodR. We sliceSinto thin, vertical strips, and model these as weights onRwith:•positionyonR, whereyis the centre of mass of the strip, and•mass inRequal to the area of the strip inS.(a) IfSis the sheet shown below, slice it into a number of vertical pieces of equal length,76
APPLICATIONS OFINTEGRATION2.3 CENTRE OFMASS ANDTORQUEapproximated by rectangles. For each rectangle, mark its centre of mass. SketchRasa rod with the same vertical height, with weights corresponding to the slices youmade ofS.