A uniform sphere has mass M 0 and radius r . A spherical cavity (no mass) of rad

Question

A uniform sphere has mass M0 and radius r. A spherical cavity (no mass) of radius r/2 is then carved within this sphere as shown in the figure(Figure 1)(the cavity's surface passes through the sphere's center and just touches the sphere's outer surface). The centers of the original sphere and the cavity lie on a straight line, which defines the xaxis. With what gravitational force will the hollowed-out sphere attract a point mass m which lies on the x axis a distance d from the sphere's center? [Hint: Subtract the effect of the "small" sphere (the cavity) from that of the larger entire sphere.]

Explanation / Answer

We know that,

density = mass/volume

rho = M0/(4/3 pi r^3)

the gone mass is:

M = rho x 4/3 pi (r/2)^3 = M0/8

We know that,

F = G m1 m2/r^2

let r = d in this case

F = -G M0m/d^2 - (- GM0/8)m/ (d - r/2)^2)

G = -G M0 m [ 1/d^2 - 1/(8 (d - r/2)^2)]

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