Motion Of The Centre Of Mass: Derivation and Formula

Motion of Centre of Mass

Edited By Vishal kumar | Updated on Jul 02, 2025 05:43 PM IST

Assume for the moment that the place at which the mass of a body or system of bodies is meant to be concentrated during its entire motion is known as the centre of mass. Another way to put it is that the mean location of mass distribution in space where the force is typically applied is the average position of all the system components, leading to a linear acceleration devoid of any rotational acceleration. The body's or system's condition of motion or rest will not change if an external force is applied to the centre of mass of the body or system of bodies.

This Story also Contains

Velocity of the Centre of Mass
Solved Examples Based On Motion of the Centre of Mass
Summary

Motion of Centre of Mass

In this article, we will cover the concept of velocity of the centre of mass and acceleration of the centre of mass under the topic of motion of the centre of mass. This topic falls under the broader category of Rotational Motion, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), more than five questions have been asked on this concept. It's also an important topic from NEET's point of view.

Let's read this entire article to gain an in-depth understanding of the motion of the centre of mass.

Velocity of the Centre of Mass

$
\vec{v}_{C M}=\frac{m_1 \overrightarrow{v_1}+m_2 \vec{v}_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots}
$
where: $\mathrm{m}_1, \mathrm{~m}_2 \ldots$ - - are the mass of all the particles and $\overrightarrow{v_1}, \overrightarrow{v_2} \ldots \ldots$ are velocities of all the particles.

Similarly momentum of the system $=P_{\text {sys }}=M v_{c m}$

Acceleration of Centre of Mass

$
\vec{a}_{C M}=\frac{m_1 \overrightarrow{a_1}+m_2 \vec{a}_2 \ldots \ldots .}{m_1+m_2 \ldots \ldots}
$
$\mathrm{m}_1, \mathrm{~m}_2$ are the mass of all the particles $\overrightarrow{a_1}, \overrightarrow{a_2} \cdots$ are their respective acceleration.

Similarly Net force on the system $=F_{n e t}=M a_{c m}$
And $F_{n e t}=\overrightarrow{F_{e x t}}+\overrightarrow{F_{i n t}}$

And we know that both the action and reaction of an internal force must be within the system. In this way, vector summation will cancel all internal forces and hence net internal force on the system is zero.

So $\overrightarrow{F_{n e t}}=M \overrightarrow{a_{c m}}$
If the External Force $=0$
$\vec{F}_{e x t}=0 \Rightarrow M \vec{a}_{c m}=0 \Rightarrow \vec{a}_{c m}=0$
if $\vec{a}_{c m}=0 \Rightarrow v_{c m}=$ constant
If $v_{c m}=$ constant $\Rightarrow P_{\text {sys }}=$ constant

So it implies that the total momentum of the system must remain constant.

i.e. if no external force is acting on the system, the net momentum of the system remains constant. This is nothing but the principle of conservation of momentum in the absence of external forces. Which says ìf the resultant external force is zero on the system, then the net momentum of the system must remain constant.

Special Case

If External Force = 0 and Velocity of Centre of Mass = 0

The centre of mass remains at rest. Individual components of a system may move and have non-zero momentum due to mutual forces but the net momentum of the system remains zero.

Solved Examples Based On Motion of the Centre of Mass

Example 1: identical particles move towards each other with velocities 2v and v respectively. The velocity of the centre of mass is :

1) $\nu$
2) $\nu / 3$
3) $\nu / 2$
4) zero

Solution:

$\begin{aligned} & v_c=\frac{m_1 v_1+m_2 v_2}{m_1+m_2} \\ & o r \quad v_c=\frac{m(2 v)+m(-v)}{m+m}=\frac{v}{2}\end{aligned}$

Hence, the answer is option (3).

Example 2: The arrangement of two masses $m_1$ and $m_2$ are as shown in the fig Assume strings and pulley are massless and frictionless. Find the acceleration of the centre of mass of the system let $m_1=3 \mathrm{Kg}$ and $m_2=1 \mathrm{Kg}$.

1) g/2

2) g/3

3) g/4

4) g

Solution:

Acceleration of centre of mass -
$
\vec{a}_{C M}=\frac{m_1 \vec{a}_1+m_2 \vec{a}_2 \ldots \ldots \cdots}{m_1+m_2 \ldots \ldots .}
$
- wherein $\mathrm{m}_1, \mathrm{~m}_2$ are the mass of all the particles $\overrightarrow{a_1}, \overrightarrow{a_2} \cdots$ are their respective acceleration.
acceleration of system
$
a=\frac{\left(m_1-m_2\right)}{\left(m_1+m_2\right)} g
$
$\vec{a}_{c m}=\frac{\left(m_1 \vec{a}_1+m_2 \vec{a}_2\right)}{\left(m_1+m_2\right)}$
$
\vec{a}_{c m}=\frac{\left(m_1(-a)+m_2 a\right)}{\left(m_1+m_2\right)} \Rightarrow-\left(\frac{m_1-m_2}{m_1+m_2}\right) a
$

The magnitude of acceleration of the centre of mass is
$
a_{c m}=\left(\frac{m_1-m_2}{m_1+m_2}\right) a
$

$\begin{aligned} & a=\frac{(3-1)^2}{(3+1)^2} g \\ a= & g / 4\end{aligned}$

Example 3: A body A of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $\frac{1}{3} M$ and another body $C$ of mass $\frac{2}{3} M$. The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of the body $A$ towards

1) body $C$

2) body B

3) depends on the height of the breaking

4) does not shift

Solution:

If External Force = 0 and Velocity of Centre of Mass = 0

The centre of mass remains at rest. Individual components of a system may move and have non-zero momentum due to mutual forces but the net momentum of the system remains zero.

The centre of mass of bodies B and C taken together does not shift as no external force is applied horizontally.

Hence, the answer is option (4).

Example 4: A man of mass 80 Kg stands on a plank of mass 40 Kg. The plank is lying on a smooth horizontal floor. Initially, both are at rest. The man starts walking on the plank towards the north and stops after moving a distance of 6m on the plank. The plank will move

1) 6m, south

2) 4m, north

3) 4m, south

4) 6m, north

Solution:

As we have learned

If External Force = 0 -

$\vec{P}_{\text {total }}=\text { constant }$

wherein

i.e. if no external force is acting on the system, the net momentum of the system remains constant.

\begin{aligned}
& F_{\text {ext }}=0 \\
& \text { so } \vec{a}_{c m}=0 \quad \vec{v}_{c m}=\text { cost }=0 \\
& \vec{v}_{c m}=\frac{m_1 \vec{v}_1+m_2 \vec{v}_2}{m_1+m_2}=0 \\
& \text { or } m_1 v_1+m_2 v_2=0 \\
& m_1 \frac{d x}{d t}+m_2 \frac{d x_2}{d t}=0 \\
& \text { or } m_1 \vec{x}_1+m_2 \vec{x}_2=0
\end{aligned}

\begin{aligned}
& \quad x_1=(6-x) \\
& x_2=-x \\
& m_1(6-x)-m_2 x=0 \\
& 80(6-x)=40 x \\
& x=4 m, \text { south }
\end{aligned}

Example 5: A projectile of mass M is fired so that the horizontal range is 4 Km. At the highest point, the projectile explodes in two parts of masses M/4 and 3M/4 respectively and the heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from the point of firing )of the lighter part is :

1) 16 Km

2) 1 Km

3) 10 Km

4) 2 Km

Solution:

\begin{aligned}
& X_{C O M}=\frac{m_1 x_1+m_2 x_2}{m_1+m_2} \\
& R=\frac{\frac{M}{4} x+\frac{3 M}{4} \times \frac{R}{2}}{M} \\
& \Rightarrow x=10 \mathrm{Km}
\end{aligned}

Hence, the answer is the option 3.

Summary

The forces that the system's particles apply to one another are known as internal forces; but, according to Newton's third rule, these internal forces only exist in pairs that are equal in magnitude and directed in opposing directions. Thus, they have a zero net sum.

Newton’s third law states that for every action, there is an equal and opposite reaction; therefore, the gravitational forces between them are identical in magnitude but opposite in direction.

Frequently Asked Questions (FAQs)

1. What happens to the center of mass when an object rotates?

For a rigid body, the position of the center of mass relative to the body remains fixed during rotation. However, the absolute position of the center of mass in space can change as the object rotates and moves.

2. What is the significance of the center of mass in projectile motion?

In projectile motion, the center of mass follows a parabolic path under the influence of gravity, regardless of any rotation or change in shape of the object. This simplifies the analysis of complex projectile motions.

3. How is the center of mass calculated for a system of discrete particles?

For a system of discrete particles, the center of mass is calculated by multiplying each particle's mass by its position, summing these products, and then dividing by the total mass of the system.

4. What is the center of mass of a system of particles?

The center of mass is the average position of mass in a system of particles. It's a unique point that behaves as if all the system's mass were concentrated there, even if no actual particle is located at that point.

5. How does the concept of center of mass apply to the human body?

The center of mass of the human body is a crucial concept in biomechanics. It's typically located around the navel area but can shift with body position and movement, affecting balance and stability.

6. What is the relationship between the center of mass and the moment of inertia tensor?

The moment of inertia tensor, which describes an object's resistance to rotational acceleration, is typically defined with respect to the center of mass. This choice simplifies many calculations in rigid body dynamics.

7. How does the motion of the center of mass relate to Newton's laws?

The center of mass of a system moves as if all the external forces acting on the system were applied to a single particle of mass equal to the total mass of the system, located at the center of mass. This is a direct application of Newton's second law to the system as a whole.

8. How does the motion of the center of mass relate to conservation of momentum?

The motion of the center of mass is directly linked to the system's total momentum. If no external forces act on a closed system, the center of mass moves with constant velocity, reflecting the conservation of linear momentum.

9. How does the concept of center of mass apply to molecular physics?

In molecular physics, the center of mass is used to separate the motion of molecules into translational motion of the center of mass and rotational and vibrational motions about the center of mass, simplifying complex molecular dynamics.

10. Can the center of mass of a system accelerate if no external forces are present?

No, if no external forces are acting on a closed system, the center of mass cannot accelerate. It will either remain at rest or move with constant velocity, in accordance with Newton's first law.

11. What is meant by the term "center of mass frame"?

The center of mass frame is a reference frame where the center of mass of a system is at rest. In this frame, the total momentum of the system is zero, which often simplifies the analysis of complex motions and collisions.

12. Can the center of mass be located outside the physical boundaries of an object?

Yes, the center of mass can be located outside the physical boundaries of an object. For example, in a donut-shaped object (torus), the center of mass is located in the empty space at the center.

13. How does the distribution of mass affect the location of the center of mass?

The center of mass is influenced by how mass is distributed within a system. If mass is evenly distributed, the center of mass will be at the geometric center. If mass is concentrated in one area, the center of mass will shift towards that area.

14. What is the relationship between center of mass and center of gravity?

In a uniform gravitational field, the center of mass and center of gravity coincide. However, in non-uniform fields (like near the Earth's surface), they can be slightly different, though this difference is usually negligible for most practical purposes.

15. How does adding or removing mass affect the position of the center of mass?

Adding or removing mass from a system will shift the center of mass towards or away from the point where mass is added or removed, respectively. The magnitude of the shift depends on the amount of mass changed and its distance from the original center of mass.

16. What is the relationship between the center of mass and rotational inertia?

The rotational inertia of an object depends on the mass distribution relative to the axis of rotation. The parallel axis theorem relates the rotational inertia about any axis to the rotational inertia about an axis through the center of mass.

17. What is meant by the term "barycenter" and how does it relate to the center of mass?

The barycenter is the center of mass of two or more bodies that orbit each other. In the context of celestial mechanics, it's the point around which celestial bodies orbit. For example, the Earth and Moon orbit around their common barycenter.

18. What is the role of the center of mass in understanding the stability of structures?

The position of the center of mass is crucial for structural stability. A lower center of mass generally increases stability. Engineers consider this when designing buildings, vehicles, and other structures to ensure they remain upright under various conditions.

19. How does the concept of center of mass apply to quantum mechanics?

In quantum mechanics, the center of mass coordinate is used to separate the internal motion of a system (like an atom or molecule) from its overall translational motion. This separation simplifies the quantum mechanical treatment of many-body systems.

20. What is the relationship between the center of mass and the equilibrium of a body?

For a body to be in equilibrium, the sum of all torques about any point must be zero. Often, it's most convenient to consider torques about the center of mass, as the gravitational torque about this point is always zero.

21. How does the concept of center of mass apply to the field of materials science?

In materials science, the center of mass concept is used in analyzing the behavior of composite materials, the dynamics of crystal lattices, and in understanding how materials respond to stress and strain at both microscopic and macroscopic levels.

22. How does the concept of center of mass apply to the field of nanotechnology?

In nanotechnology, understanding the center of mass is crucial for manipulating and controlling nanoscale objects. It's particularly important in designing nanorobots and in understanding the behavior of complex molecular machines.

23. Why is the concept of center of mass important in astrophysics?

In astrophysics, the center of mass concept is crucial for understanding the motion of celestial bodies. For example, planets orbit around the center of mass of the solar system, not exactly around the Sun's center.

24. How is the center of mass concept used in rocket propulsion?

In rocket propulsion, the changing mass distribution due to fuel consumption causes the center of mass to shift. This shift must be accounted for in the rocket's design and control systems to maintain stability and desired trajectory.

25. What is the significance of the center of mass in analyzing collisions?

The center of mass is crucial in collision analysis as it allows us to separate the motion of the system as a whole from the internal motions of the colliding objects. The behavior of the center of mass follows simple rules even in complex collisions.

26. How does the concept of center of mass apply to extended objects in gravitational fields?

For extended objects in gravitational fields, the gravitational force can be considered to act at the center of mass. This simplification allows us to treat complex objects as point masses in many gravitational calculations.

27. How does the motion of the center of mass relate to energy conservation?

The kinetic energy of a system can be separated into the kinetic energy of the center of mass motion and the kinetic energy of motion relative to the center of mass. This separation is useful in applying energy conservation principles to complex systems.

28. How does the center of mass behave in a system undergoing an explosion?

In an explosion, while individual parts may scatter in various directions, the center of mass of the entire system continues to move as if no explosion occurred, assuming no external forces. This illustrates the conservation of momentum principle.

29. What is the significance of the center of mass in sports biomechanics?

In sports biomechanics, understanding the center of mass is crucial for analyzing and improving athletic performance. It helps in studying balance, stability, and efficient movement in activities like gymnastics, diving, and track and field events.

30. How does the concept of center of mass apply to fluid dynamics?

In fluid dynamics, the center of mass concept is used to analyze the bulk motion of fluids. It's particularly useful in studying the motion of fluid bodies like water droplets or air masses in meteorology.

31. How does the motion of the center of mass differ between elastic and inelastic collisions?

The motion of the center of mass remains unchanged by internal forces during both elastic and inelastic collisions. However, while the relative motion of colliding objects changes in both cases, only in elastic collisions is kinetic energy conserved relative to the center of mass.

32. What is the significance of the center of mass in analyzing the motion of articulated bodies?

For articulated bodies like robots or the human body, the center of mass changes position relative to the body as joints move. Understanding this is crucial for maintaining balance, planning movements, and analyzing gait in biomechanics and robotics.

33. How does the motion of the center of mass relate to the work-energy theorem?

The work-energy theorem can be applied separately to the motion of the center of mass and to motion relative to the center of mass. The total work done by external forces equals the change in kinetic energy of the center of mass plus the change in internal energy of the system.

34. What is the significance of the center of mass in analyzing the motion of galaxies?

In galactic dynamics, the center of mass concept is used to study the overall motion of galaxies and galaxy clusters. It helps in understanding galactic rotation, interactions between galaxies, and the large-scale structure of the universe.

35. How does the concept of center of mass apply to nuclear physics?

In nuclear physics, the center of mass frame is often used to simplify calculations of nuclear reactions and decays. It's particularly useful in analyzing collisions between nuclei and in understanding the dynamics of nuclear systems.

36. What is the role of the center of mass in understanding the precession of spinning objects?

The precession of a spinning object, like a gyroscope, can be understood by considering the torque acting on the object about its center of mass. The motion of the center of mass itself is not affected by the precession, illustrating the separation of translational and rotational motions.

37. How does the distribution of mass in the Earth affect its center of mass?

The Earth's center of mass is slightly offset from its geometric center due to variations in density and mass distribution within the planet. This offset affects the Earth's rotation and orbital dynamics, and is important in geophysics and satellite orbit calculations.

38. What is the significance of the center of mass in analyzing the motion of deformable bodies?

For deformable bodies, while the shape may change, the motion of the center of mass remains simple if no external forces are present. This principle is useful in analyzing complex motions like those of a cat falling and landing on its feet.

39. How does the concept of center of mass apply to particle physics experiments?

In particle physics, collisions are often analyzed in the center of mass frame. This frame simplifies the analysis of particle creation and decay processes, and is crucial for interpreting data from particle accelerators.

40. How does the motion of the center of mass relate to the virial theorem in physics?

The virial theorem, which relates the average kinetic energy of a stable system to its average potential energy, is often applied in the center of mass frame. This application is particularly useful in astrophysics for understanding the stability of star clusters and galaxies.

41. What is the significance of the center of mass in analyzing the motion of tethered satellites?

In the analysis of tethered satellite systems, the motion of the center of mass of the entire system (including the tether) follows a simple orbit, while the relative motion of the satellites about the center of mass can be complex and is influenced by the tether forces.

42. How does the concept of center of mass apply to the field of robotics?

In robotics, tracking and controlling the center of mass is crucial for maintaining balance, especially in bipedal robots. The center of mass trajectory is a key factor in planning stable walking gaits and other complex movements.

43. What is the role of the center of mass in understanding the formation of planetary systems?

The center of mass concept is essential in modeling the formation and evolution of planetary systems. It helps explain how planets and other bodies coalesce from rotating disks of gas and dust, and how they settle into stable orbits.

44. How does the motion of the center of mass relate to the concept of pseudoforces in non-inertial frames?

In non-inertial frames, like a rotating reference frame, pseudoforces appear to act on objects. These forces can be understood by considering the motion of the center of mass in an inertial frame and then transforming to the non-inertial frame.

45. What is the significance of the center of mass in analyzing the motion of asteroids and comets?

The center of mass motion of asteroids and comets determines their orbits around the Sun. However, their rotation about their center of mass, which can be complex due to irregular shapes, affects phenomena like outgassing in comets and the Yarkovsky effect in asteroids.

46. What is the role of the center of mass in understanding the dynamics of black hole mergers?

In the merging of black holes, the center of mass of the system emits gravitational waves. The motion of this center of mass, along with the complex orbital dynamics of the merging black holes, is crucial for predicting and interpreting gravitational wave signals.

47. How does the motion of the center of mass relate to the concept of reduced mass in two-body problems?

The reduced mass is a mathematical construct that allows a two-body problem to be treated as a one-body problem relative to the center of mass. This simplification is widely used in classical mechanics, quantum mechanics, and astrophysics.

48. What is the significance of the center of mass in analyzing the motion of weather systems?

In meteorology, the center of mass concept is used to track and predict the movement of large-scale weather systems like hurricanes. The overall motion of these systems can often be approximated by the motion of their center of mass.

49. What is the role of the center of mass in understanding the dynamics of star clusters?

The center of mass of a star cluster moves through the galaxy under the influence of galactic tides and other external forces. Internally, stars orbit around this common center of mass, with their collective behavior determining the cluster's stability and evolution.

50. How does the motion of the center of mass relate to the concept of normal modes in vibrating systems?

In systems with multiple vibrating components, like molecules or coupled oscillators, the motion can be decomposed into normal modes. The center of mass motion often corresponds to one of these modes, typically the lowest frequency mode representing the overall translation of the system.