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

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

**Contents in this article:**

**Definition of acceleration center of Mass****Acceleration of center of mass equation****How to find acceleration of center of mass?****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. For a system moving in straight line with constant velocity have 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 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 **m _{1}, m_{2}, m_{3}, ……, m_{n}** moving with uniform accelerations

**a**respectively. Then the formula for acceleration center of mass of the discrete mass system is,

_{1}, a_{2}, a_{3}, …., a_{n}\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 entire body.

**How to find Acceleration center of mass?**

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

There is an example at below that shows the calculation of center of mass acceleration using 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/s ^{2} to the 6 kg mass in the direction of 3 kg mass. Find the acceleration of the center of the mass.**

**Answer:**

This is an example of two-body discrete system. The masses are **m _{1} = 6 kg** and

**m**and their accelerations are

_{2}= 3 kg**a**and

_{1}= 10 m/s**a**respectively. Then using equation-(1) we get,

_{2}= 0 m/sthe 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/s^{2}. (**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.

Thanks for reading this article and keep visiting this website.

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