Problem 2B-07*: Centers of mass of a spherical shell and a sphere
(a) Show that the center of mass of a uniform thin spherical shell is its center point.
(b) Show that the center of mass of a uniform sphere is its center point.
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(a) Let the center of a uniform thin spherical shell be denoted as point O. Any
point A on the surface of the sphere can be mapped with regard to the center
point O to a point A' on the spherical surface of the other side. We will have
AO = OA'. If A is an area, instead of a point, it can be mapped to a congruent
area A' on the surface of the sphere of the other side as shown in Fig. 1.
Let us cut the southern sphere of the surface of the spherical shell into N pieces and label them from 1 to N. The k-th piece can be mapped to a congruent piece on the northern sphere. We label this mapped piece the (-k)-th piece. The center of mass of the k-th piece is denoted as Ck, and the center of mass of the (-k)-th piece as C-k. The position vector OCk is denoted as rk, and the position vector OC-k is denoted as r-k. The masses of k-th piece and (-k)-th piece should be same, so they are denoted as mk. Since we must have
rk + r-k = 0,
we can write
mk(rk + r-k) = 0.
Thus we have
m1(r1 + r-1) + m2(r2 + r-2) + · · · + mN(rN + r-N) = 0. (1)
Eq. (1) is equivalent to Eq. (4) of Topic 2B-02, so the center point O is the center of mass of the thin uniform spherical shell.
(b) Let the center of the uniform sphere be denoted as point O. A uniform sphere can be considered to consist of a large number of thin uniform spherical shells each of that shares point O as its center of mass. The position vector from point O to the center of mass of the k-th shell, rk, is zero since the center of mass coincides with point O. Thus Eq. (4) of Topic 2B-02 is automatically satisfied. Therefore, point O is the center of mass of the uniform sphere.