Dr. Mitchell A. Hoselton
Halliday, Resnick and Walker
Fundamentals of Physics
AP Physics C
Serway and Beichner
Physics for Scientists and Engineers
Page 1
Center of Mass - V
P-V. A solid spherical metal ball of radius 2R has had a smaller sphere of radius R removed from it.
The metal in the sphere is of uniform density. Find the Center of Mass of the larger sphere after the
smaller sphere has been scooped out.
If the smaller sphere were present, the center of mass
would be at the center of the large sphere. Take the
center of the large sphere as the origin and find the new
center of mass relative to that origin.
The position of the center of mass of the smaller sphere is
known to be –R.
The exercise on the right shows the derivation
of the expression for x2, the center of mass of
the larger sphere with the smaller sphere
scooped out of it. Unfortunately, we do not
know the masses. We will exploit the uniform density of the metal to convert this equation in terms of
mass to an equation in terms of volumes. For this we need to use the density.
First, we recall that the density is defined as:
This allows us to rewrite the
expression for the new center
of mass, x2, as follows: