EXAMPLE 3 The density at any point on a semicircular lamina is proportional to the distance from the center of the circle. Find the center of mass of the lamina. + y² = a° SOLUTION Let's place the lamina as the upper half of the circle x? + y? = a?. (See the figure.) Then the distance from a point (x, y) to the center of the circle (the origin) is V? + y?. Therefore the density function is (0, ) D p(x, y) = K/x? + y? Video Example ) where K is some constant. Both the density function and the shape of the lamina suggest that we convert to polar coordinates. Then Vx + y² = r and the region D is given by osrsa, oses n. Thus the mass of the lamina is p(x, y) dA m - dA dr de K 2 dr o Jo Jo Both the lamina and the density function are symmetric with respect to the y-axis, so the center of mass must lie on the y-axis, that is x = [ is given by 1. The y-coordinate 3 Kra r sin(e) (Kr)r dr de ma3 sin(@) de or dr cos(@ The center of mass is located at the point (x, y) =

EXAMPLE 3 The density at any point on a semicircular lamina is proportional to the distance from the center of the circle. Find the center of mass of the lamina. + y² = a° SOLUTION Let's place the lamina as the upper half of the circle x? + y? = a?. (See the figure.) Then the distance from a point (x, y) to the center of the circle (the origin) is V? + y?. Therefore the density function is (0, ) D p(x, y) = K/x? + y? Video Example ) where K is some constant. Both the density function and the shape of the lamina suggest that we convert to polar coordinates. Then Vx + y² = r and the region D is given by osrsa, oses n. Thus the mass of the lamina is p(x, y) dA m - dA dr de K 2 dr o Jo Jo Both the lamina and the density function are symmetric with respect to the y-axis, so the center of mass must lie on the y-axis, that is x = [ is given by 1. The y-coordinate 3 Kra r sin(e) (Kr)r dr de ma3 sin(@) de or dr cos(@ The center of mass is located at the point (x, y) =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.2: Properties Of Division

Problem 53E

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