Question

Three uniform spheres of radius 2R, R, and 3R are placed in a line, in the order given, so their centers are lined up and the spheres are touching each other (i.e. the 2R sphere is on the left, the 3R sphere is on the right). How far is the center of mass of the system from the center of mass of the smallest sphere

Answer 1

Answer:

x = 2.33 R from the center of mass of the smallest sphere.

Explanation:

Due to the symmetry of the spheres, the center of mass of any of them, is located just in the center of the sphere.

If we align the centers of the spheres with the x-axis, the center of mass of any of them will have only coordinates on the x-axis, so the center of  mass of the system will have coordinates on the x-axis only also.

By definition, the x-coordinate of the center of mass of a set of discrete masses m₁, m₂, m₃, can be calculated as follows:

$Xcm = \frac{m1*x1+m2*x2+m3*x3}{m1+m2+3}$

In this case, we need to get the coordinates of the center of mass of each sphere:

If we place the spheres in such a way that the center of the first sphere has the x-coordinate equal to its radius (so it is just touching the origin), we will have:

x₁ = 2*R

For the second sphere, the center will be located at a distance equal to the diameter of  the first sphere plus its own radius, as follows:

x₂ = 4*R + R = 5*R

Finally, for the third sphere, the center will be located at a distance equal to the diameter of  the first sphere, plus the diameter of the second sphere,  plus its own radius, as follows:

x₃ = 4*R + 2*R + 3*R = 9*R

We can calculate the mass of each sphere (assuming that all are from the same material, with a constant density), as the product of the density and the volume:

m = ρ*V

For a sphere, the volume can be calculated as follows:

$\frac{4}{3} *\pi *(r)^{3}$

So, we can calculate the masses of the spheres, as follows:

m₁ = ρ*$\frac{4}{3} *\pi *(2r)^{3}$

m₂ = ρ*$\frac{4}{3} *\pi *(r)^{3}$

m₃ = ρ*$\frac{4}{3} *\pi *(3r)^{3}$

The total mass can be calculated as follows:

M= ρ*$\frac{4}{3} *\pi$ * (8*r³ + r³ + 27*r³) =ρ*$\frac{4}{3} *\pi$ * 36*r³

Replacing by the values, and simplifying common terms, we can calculate the x-coordinate of the center of mass of the system as follows:

$Xcm = \frac{m1*x1+m2*x2+m3*x3}{m1+m2+3}$

$Xcm = \frac{(8*R^{3} *2*R)+(R^{3}*(5*R))+27*R^{3}*(9*R))}{36*R^{3} }=\frac{264*R^{4} x}{36*R^{3}} = 7.33 R$

As the x-coordinate of the center fof mass of the entire system is located at 7.33*R from the origin, and the center of mass of the smallest sphere is located at 5*R from the origin, the center of mass of the system is located at a distance d:

d = 7.33*R - 5*R = 2.33 R