Question

A uniform solid sphere of radius $R$ has a spherical hole or radius $a$ in it. Find the position of its center of mass.

• (A) $x_{cm}=-a^{3}b/(R^3-a^3),y_{cm}=0,z_{cm}=0$
• (B) $x_{cm}=a^{3}b/(R^3-a^3),y_{cm}=0,z_{cm}=0$
• (C) $x_{cm}=b^{3}a/(R^3-a^3),y_{cm}=0,z_{cm}=0$
• (D) None of these

Answer 1

The Correct Option is A

Explanation:
Imagine the total hole filled with matter so as to produce uniform sphere of radius $\mathrm{R}$ and density $\rho$. The filled hole can then be represented by point $\frac{4}{3}\pi a^3\rho$ at $(b,0,0)$ The remained of sphere of mass $\frac{4}{3}\pi (R^3-a^3)\rho$ at $(x_{cm},0,0)$ The centre of mass of these two-part must be at centre of the mass of the sphere So $\frac{4}{3}\pi (R^3-a^3)\rho x_{cm}+\frac{4}{3}\pi a^3\rho b=0$$x_{cm}=\frac{-a^{3}b}{R^3-a^3}$Hence, the correct option is (A).

Answer 2

The center of mass of a body is a point where the whole mass of the body is supposed to be concentrated.

• It is the mean location of a distribution of mass in space. 
• A rigid body having uniform density, its center of mass is located at the centroid.
• For instance, the center of mass of a circular plate would be at its center.
• Centre of mass may or may not fall within the object.

Calculation of Center of Mass

The two main methods to determine the center of mass of an object as described below:

Table Edge Method

• This method is used if the rigid body has at least one flat surface.
• The object is moved gently towards the edge of a table without rotating the surface.
• At the moment when the object appears to be about to fall, a line is drawn parallel to the edge of the table.
• Rotating the object by 90° and the same process is repeated.
• The point of intersection of the two lines gives the center of mass.

Plumb Line Method

• This method is applied to objects that can be freely suspended about a point of rotation.
• A piece of cardboard with an irregular shape suspended on a pinboard is an example of this method.
• Under gravity, the cardboard rotates freely around the pin until it reaches a stable position.
• Then, a plumb line is hung from the pin and is used to mark a line on the object.
• The pin is then moved to another location and the procedure is repeated again.
• The center of mass then lies beneath the point of intersection of these two lines.