Chapter 15, Problem 10PP

(a) A lamina has constant density ρ and takes the shape of a disk with center the origin and radius R. Use Newton’s Law of Gravitation (see Section 13.4) to show that the magnitude of the force of attraction that the lamina exerts on a body with mass m located at the point (0, 0, d) on the positive z-axis is $F=2\pi Gm\rho d\left(\frac{1}{d}-\frac{1}{\sqrt{R^2+d^2}}\right)$

[Hint: Divide the disk as in Figure 15.3.4 and first compute the vertical component of the force exerted by the polar subrectangle R_ij.]

(b) Show that the magnitude of the force of attraction of a lamina with density ρ that occupies an entire plane on an object with mass m located at a distance d from the plane is $F=2\pi Gm\rho$ Notice that this expression does not depend on d.