Acceleration & Velocity of Centre of Mass (COM)

The center of mass (COM) of a system is a point that behaves as if all the mass of the system were concentrated there and all external forces were applied at this point. Understanding the motion of the center of mass is crucial in physics, as it simplifies the analysis of the system's dynamics.

Velocity of Centre of Mass

The velocity of the center of mass (( \vec{V}_{COM} )) is the rate of change of its position with time. It is a vector quantity and is given by the total momentum of the system divided by the total mass of the system.

The formula for the velocity of the center of mass is:

[ \vec{V}_{COM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} ]

where ( m_i ) is the mass of the ( i^{th} ) particle and ( \vec{v}_i ) is the velocity of the ( i^{th} ) particle.

Acceleration of Centre of Mass

The acceleration of the center of mass (( \vec{a}_{COM} )) is the rate of change of its velocity with time. It is also a vector quantity and is given by the total external force acting on the system divided by the total mass of the system.

The formula for the acceleration of the center of mass is:

[ \vec{a}{COM} = \frac{\sum \vec{F}{ext}}{\sum m_i} ]

where ( \vec{F}_{ext} ) is the external force acting on the system.

Table: Differences and Important Points

Property	Velocity of COM	Acceleration of COM
Definition	Rate of change of position of COM with time	Rate of change of velocity of COM with time
Formula	( \vec{V}_{COM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} )	( \vec{a}{COM} = \frac{\sum \vec{F}{ext}}{\sum m_i} )
Depends on	Masses and velocities of individual particles	External forces acting on the system
SI Unit	meters per second (m/s)	meters per second squared (m/s²)
Direction	Direction of COM's motion	Direction of net external force

Examples

Example 1: Velocity of COM

Consider a system of two particles with masses ( m_1 = 2 ) kg and ( m_2 = 3 ) kg, moving with velocities ( \vec{v}_1 = 4 ) m/s and ( \vec{v}_2 = 2 ) m/s, respectively. The velocity of the COM is:

[ \vec{V}_{COM} = \frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2} = \frac{2 \times 4 + 3 \times 2}{2 + 3} = \frac{14}{5} = 2.8 \text{ m/s} ]

Example 2: Acceleration of COM

Assume the same system of two particles is subjected to external forces ( \vec{F}_1 = 6 ) N and ( \vec{F}_2 = 4 ) N. The acceleration of the COM is:

[ \vec{a}_{COM} = \frac{\vec{F}_1 + \vec{F}_2}{m_1 + m_2} = \frac{6 + 4}{2 + 3} = \frac{10}{5} = 2 \text{ m/s}^2 ]

Conclusion

The concepts of velocity and acceleration of the center of mass are fundamental in understanding the motion of a system of particles. They allow us to predict the behavior of the system under the influence of external forces and simplify the analysis of complex systems by focusing on the motion of a single point, the COM.