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Physics Formulas

Center of Mass Formula

Before to start, let us get some terms out of the way. Also, we need to know the center of mass for an object or a set of objects. It is the point about which the entire mass of the system is equally distributed. In this regard, the center of gravity is also important to know. It refers to the point about which the forces of gravity are balanced. In physics center of mass of an object is very important to find accurately. In this article, we will discuss the center of mass formula with examples. Let us begin learning!

Center of mass formula

Center of Mass Formula

What is the Center of Mass?

The center of mass of a body or system of a particle is defined as a point where the whole of the mass of the body or all the masses of a set of particles appeared to be concentrated.

When we are studying the dynamics of the motion of the system of a particle as a whole, then we need not worry about the dynamics of individual particles of the system. Rather than we have to focus on the dynamic of a unique point corresponding to that system.

The motion of this unique point is similar to the motion of a single particle whose mass is equal to the sum of all individual particles of the system. Also the resultant of all the forces exerted on all the particles of the system by surrounding bodies is exerted directly to that particle. This point is known as the center of mass of the system of particles.

Centre of Mass Formula

We can extend the formula for the center of mass to a system of particles. We can apply the equation individually to each axis also.

Although the center of mass and the center of gravity often coincide, these are all different concepts. Meanwhile, the center of gravity and the center of mass are only equal when the entire system is subject to a uniform gravitational field.

\( x_{com} = \frac{\sum_{n}^{i=0}m_ix_i}{M}\)

\(y_{com} = \frac{\sum_{n}^{i=0}m_iy_i}{M}\)

\( z_{com} = \frac{\sum_{n}^{i=0}m_iz_i}{M} \)

We can use the above formula if we have pointed objects. On the other hand, we have to take a different approach if we have to find the center of mass of an extended object like a rod. Then we will consider a differential mass and its position and then integrate it over the entire length.

\( x_{com} = \frac{\int xdm}{M}\)

\( y_{com} = \frac{\int ydm}{M}\)

\(z_{com} = \frac{\int zdm}{M}\)

Where,

\(x_{com} , y_{com} and z_{com}\) Center of mass along x, y, and z-axis
M The total mass of system
n Number of objects
\(m_i\) Mass of the ith object
\(x_i\) Distance from the x-axis of ith object

Solved Examples

Q.1: The minute hand of a clock consists of an arrow with a circle connected by a piece of metal with almost zero mass. The mass of the arrow is 15.0 g. The circle has a mass of 60.0 g. If the circle is at position 0.000 m, the position of an arrow is at 0.100 m, then find out the center of mass?

Solution:

The center of mass of the minute-hand:

\( x_{com} = \frac{\sum_{n}^{i=0}m_ix_i}{M}\)

\( x = \frac{\sum_{n}^{i=0}m_ix_i}{M}\)

\(x = \frac{m_1x_1+m_2x_2}{m_1+m_2}\)

= \( \frac{60 \times 0.0 + 15 \times 0.10}{60 + 15}\)

= \( \frac{1.50}{75}\)

x = 0.02 m

The center of mass will be at 0.020 m from the circle.

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