18 02A Topic 34 Applications of double integration Author Jeremy Orloff Read TB 20 3 Center of Mass m1 1 x1 In one dimension x cm m1 2 x cm x1 m2 1 x2 xcm x1 x2 2 m2 1 x2 xcm 2x1 x2 3 In general xcm weighted average of position P Pmi xi mi For continuous density x a b M Rb a x dx xcm R x dm M R R x x dx x dx In 2 dimensions RR RR RR M x y dA xcm x x y dA M ycm y x y dA M R R R Moment of inertia about a line About a line d I RR R RR 2 d2 dm d x y dA R Polar moment of inertia mom of intertia about the z axis dm dm d O Moment of inertia is always about an axis When we say the moment of inertia about a point it always means moment of inertia about the axis through the point and perpendicular to the plane of the object We call this the polar moment of inertia Example xy Find mass center of mass and polar moment of inertia O RR R1R1 O Mass M dA 0 0 xy dx dy 14 1o R R R R R 1 1 xcm M1 x dA M1 0 0 x2 y dy dx d 1 R 2 2 2 1 Inner not including M1 0 x2 y dy x 2y x2 1 O 0 1 R1 2 3 Outer 0 x2 dx x6 61 0 Symmetry ycm 23 RR 2 R1R1 2 Moment of inertia I r dA x y 2 xy dx dy 41 R 0 0 xcm 4 6 32 1 18 02A topic 34 2 Example Disk of radius a with center at a 0 x y 1 Find moment of inertia about O RR 2 RR 2 I r dA r dA R R In polar coords boundary circle r 2a cos 2 2 Limits 2 to 2 fix r 0 to 2a cos R 2 R 2a cos 2 I 2 0 r r dr d Inner r4 4 2a cos 4a4 cos4 0 Outer R 2 4a4 cos4 d a4 2 Note this agrees with R 2 3 2 cos 2 12 cos 4 2 2 2 3 sin 2 81 sin 4 2 2 a O a4 a4 32 M 23 a2 the parallel axis theorem Average Value 1 The average value of f x y with respect to area on a region R is area R Z Z f x y dA R Example What s the average distance of a point in a square from the center answer We center the square on the origin By symmetry this is the same as the average distance from the origin of the triangular region R shown in the picture In polar coordinates the distance is r and the area of the triangle is 21 average Z 4 Z 4 Z sec sec3 1 r r dr d 2 distance d 13 2 ln 2 1 1 2 0 3 0 0 1 o r sec R 1 Note xcm is the average value of x with respect to mass The geometric center has coordinates given by the average value of x and y with respect to area i e the center of mass when 1 End of topic 34 notes
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