# Acceleration of Center of Mass – equation, definition

We already know that the center of mass of a body is a point inside the body where its entire mass is situated. A moving body can have two other center of mass – Velocity of center of mass and acceleration of center of mass.

In another article, we have discussed the velocity center of mass. Here we are going to explain the acceleration of center of mass, its definition and equations.

1. Definition of acceleration center of Mass
2. Acceleration of center of mass equation
3. How to find acceleration of center of mass?
4. Calculation of acceleration center of mass with example

## Acceleration of Center of Mass definition

When a body moves, its center of mass will have a velocity and acceleration (if the motion is an accelerated motion). This is simply the acceleration of center of mass.

It has the same unit and dimension that of normal acceleration. A system moving in a straight line with constant velocity has zero acceleration of center of mass.

## Acceleration of center of mass equation

The formula of acceleration center of mass depends on the system of masses – whether it is a continuous or discrete system. Its equations are as followings –

### Equation of Acceleration center of mass for a discrete system of mass

We consider a discrete system of masses m1, m2, m3, ……, mn moving with uniform accelerations a1, a2, a3, …., an respectively. Then the formula for acceleration center of mass of the discrete mass system is,

$\color{Blue}A_{c}=\frac{m_{1}a_{1} + m_{2}a_{2} + m_{3}a_{3} + ....}{m_{1} + m_{2} + m_{3} + ...}$………..(1)

### Formula of Acceleration center of mass for a continuous body

For a continuous body, we need to use integral equation. Let, the acceleration of an elementary mass (dm) of a body is a. This acceleration can be a constant or a function of position on the body.

Then the equation for the acceleration center of mass of the continuous body is,

$\color{Blue}A_{c}=\frac{1}{M}\int a.dm$……..(2)

Here, M is the mass of the entire body.

## How to find Acceleration center of mass?

We can calculate the Acceleration of the central point of a body by using the above two equations. Equation-(1) is for a discrete mass system and equation-(2) is for a continuous mass system.

There is an example below that shows the calculation of center of mass acceleration using the above formulae.

## Calculation of acceleration center of mass with example

Question:

Two blocks of masses 6 kg and 3 kg are connected by a spring of negligible mass and placed on a frictionless table. A force gives an acceleration of 10 m/s2 to the 6 kg mass in the direction of 3 kg mass. Find the acceleration of the center of the mass.

This is an example of two-body discrete system. The masses are m1 = 6 kg and m2 = 3 kg and their accelerations are a1 = 10 m/s and a2 = 0 m/s respectively. Then using equation-(1) we get,

the acceleration of the center of mass of the spring-block system is, $\color{Blue}A_{c}=\frac{6×10+3×0}{6+3}$

or, $\color{Blue}A_{c}=6.67$ m/s2. (Answer)

This is all from this article on center of mass acceleration. If you have any doubt on this topic you can ask me in the comment section.