Question

Two spheres A and B of masses m and 2m and radii 2R and R respectively are placed in contact as shown. Where does the center of mass (COM) of the system lie?


A. Inside A
B. Inside B
C. At the point of contact
D. None of these

Answer 1

Hint: Before we start addressing the problem let’s understand about the center of mass. It is defined as all the masses of a body which is concentrated at the center of the body and which depend on the mass and distance from the center of the object or we can say position vector.

Formula Used:
To find the center of mass of sphere, we have,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}\]
Where,
\[{m_1}\]and\[{m_2}\] are the masses of two spheres A and B.
\[{x_1}\]and \[{x_2}\] are distances from the center of spheres.

Complete step by step solution:

Image: Spheres of radius R and 2R

Consider the two spheres A and B of masses having m and 2m with the radii 2R and R that are placed in contact with one another as shown in the figure. Then we need to find where the center of the mass lies.

In order to do that we are considering the formula to find the center of mass,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}} \\ \]
Now, put the value of \[{m_1}\],\[{m_2}\],\[{x_1}\] and \[{x_2}\] using data
\[{X_{CM}} = \dfrac{{m\left( 0 \right) + 2m\left( {R + 2R} \right)}}{{m + 2m}}\]

Since the center of mass lie in the center of the first sphere which is our reference point, therefore, \[{x_1} = 0\] and \[{x_2} = R + 2R\]
\[{X_{CM}} = \dfrac{{2m\left( {3R} \right)}}{{3m}} \\ \]
\[\therefore {X_{CM}} = 2R\]
Therefore, the center of mass of a system lies at the point of contact.

Hence, option C is the correct answer.

Note:The center of mass is used to calculate the masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics etc.