MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms G Gravitational Attraction We use triple integration to calculate the gravitational attraction that a solid body V of mass M exerts on a unit point mass placed at the origin If the solid V is also a point mass then according to Newton s law of gravitation the force it exerts is given by where R is the position vector from the origin 0 to the point V and the unit vector r R IRI is its direction d 0 If however the solid body V is not a point mass we have to use integration We concentrate on finding just the k component of the gravitational attraction all our examples will have the solid body V placed symmetrically so that its pull is all in the k direction anyway To calculate this force we divide up the solid V into small pieces having volume AV and mass Am If the density function is S x y z we have for the piece containing the point 5 Y z Thinking of this small piece as being essentially a point mass at x y z the force A F it exerts on the unit mass at the origin is given by I and its k component AF is therefore Am AF G T r IRI k which in spherical coordinates becomes using 2 and the picture SAV cos 4 AF GSAV Gcos 4 p2 p2 If we sum all the contributions to the force from each of the mass elements A m and pass to the limit we get for the k component of the gravitational force cos 4 F G p 2 s d If the integral is in spherical coordinates then dV p2sin4dpd4d0 and the integral becomes G GRAVITATIONAL ATTRACTION 1 Example 1 Find the gravitational attraction of the upper half of a solid sphere of radius a centered at the origin if its density is given by 6 Jw Solution Since the solid and its density are symmetric about the z axis the force will be in the k direction and we can use 3 or 4 Since the integral is F G JdZnJdni2 Jda p sin2 cos dp dm do which evaluates easily to rrGa2 3 Example 2 Let V be the solid spherical cap obtained by slicing a solid sphere of radius a f i by a plane at a distance a from the center of the sphere Find the gravitational attraction of V on a unit point mass a t the center of the sphere Take the density to be 1 Solution To take advantage of the symmetry place the origin a t the center of the sphere and align the axis of the cap along the z axis so the flat side of the cap is parallel to the xy plane We use spherical coordinates the main problem is determining the limits of integration If we fix and 8 and let p vary we get a ray which enters V at its flat side and leaves V on its spherical side p a f i The rays which intersect V in this way are one sees from the picture Thus by 4 those for which 0 5 5 4 as which after integrating with respect to p and 8 becomes Remark Newton proved that a solid sphere of uniform density and mass M exerts the same force on an external point mass as would a point mass M placed at the center of the sphere See Problem 6a This does not however generalize to other uniform solids of mass M it is not true that the gravitational force they exert is the same as that of a point mass M a t their center of mass For if this were so a unit test mass placed on the axis between two equal point masses M and M ought to be pulled toward the midposition whereas actually it will be pulled toward the closer of the two masses Exercises Section 5C
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