Center of Mass · aiwiki.page

The center of mass is the point representing the mass-weighted average position of an object or system of objects. In classical mechanics, it provides a way to describe the overall motion of an extended body without tracking every constituent particle: for a fixed collection of particles with constant masses, its acceleration is determined by the net external force divided by the total mass. The point need not lie within the material of the object; the center of mass of a uniform hollow sphere, for example, lies in its empty interior. (openstax.org)

Mathematical definition

For $N$ particles with masses $m_i$ and position vectors $\mathbf r_i$ , measured in the same coordinate system, the center-of-mass position is

\mathbf R=\frac{1}{M}\sum_{i=1}^{N}m_i\mathbf r_i, \qquad M=\sum_{i=1}^{N}m_i.

Each coordinate is calculated separately. For example,

X=\frac{\sum_i m_i x_i}{M}.

An equivalent characterization is

\sum_i m_i(\mathbf r_i-\mathbf R)=\mathbf 0:

the first moment of the mass distribution about its center of mass vanishes. These expressions assume a finite, nonzero total mass and well-defined first moments. (openstax.org)

For two positive masses, the center lies on the line segment joining them, nearer the larger mass. If their distances from the center are $d_1$ and $d_2$ , then

m_1d_1=m_2d_2.

As an illustrative calculation, masses of $2\,\mathrm{kg}$ and $6\,\mathrm{kg}$ at $x=0$ and $x=4\,\mathrm m$ have their center of mass at $X=3\,\mathrm m$ . This follows directly from the mass-weighted coordinate formula. (openstax.org)

For a continuous distribution, summation becomes an integral:

\mathbf R=\frac{1}{M}\int\mathbf r\,dm.

If the volume density is $\rho(\mathbf r)$ , then

M=\int_V\rho(\mathbf r)\,dV, \qquad \mathbf R= \frac{\int_V\mathbf r\,\rho(\mathbf r)\,dV} {\int_V\rho(\mathbf r)\,dV}.

Thin sheets and slender wires are treated similarly, using mass per unit area or per unit length. A composite object can be analyzed by treating each component’s mass as concentrated at that component’s own center of mass. (openstax.org)

Geometry, symmetry, and the centroid

The centroid is a geometric average of position, independent of the physical mass distribution. For an object of uniform density, the appropriate geometric centroid and center of mass coincide. For nonuniform density, they generally differ: adding dense material to one side shifts the center of mass toward that side without changing the object’s external shape. (openstax.org)

Symmetry simplifies calculations when it applies to the mass distribution, not merely to the outline. A uniform rectangular plate has its center of mass at the intersection of its diagonals. A uniform circular hoop has its center at the center of the circle, even though that point contains no hoop material. Such examples show why “center” does not necessarily mean a point inside the substance of a body. (openstax.org)

Motion and external forces

For constant constituent masses, differentiation gives the center-of-mass velocity:

\mathbf V=\frac{d\mathbf R}{dt} =\frac{1}{M}\sum_i m_i\mathbf v_i.

The total momentum is therefore

\mathbf P=\sum_i m_i\mathbf v_i=M\mathbf V.

Applying Newton’s laws of motion to the particles and summing their equations yields

M\frac{d^2\mathbf R}{dt^2} =\mathbf F_{\mathrm{ext}},

when the internal forces cancel in the total force balance. The external force is the sum of forces exerted by agents outside the chosen system. (openstax.org)

Internal interactions can redistribute motion among the particles without changing the total momentum. Consequently, an isolated system’s center of mass moves at constant velocity. This remains true through a collision or explosion, although the individual particles may change velocity substantially. For example, in approximately uniform gravity and with air resistance neglected, the center of mass of all fragments from an exploding projectile continues along the trajectory dictated by gravity. (openstax.org)

The point-particle description concerns overall translation; it does not replace a description of rotation or deformation. A body can experience zero net external force while an external torque changes its rotational motion. Both force balance and torque balance are required for static equilibrium. (openstax.org)

Center-of-mass frame and kinetic energy

A reference frame moving with the center of mass is called the center-of-mass frame. In Newtonian mechanics, the total momentum in this frame is zero. It is an inertial reference frame when the center of mass moves uniformly relative to another inertial frame; a frame following an accelerating center of mass is not inertial. (ocw.mit.edu)

Writing each particle’s velocity as

\mathbf v_i=\mathbf V+\mathbf u_i,

where $\mathbf u_i$ is its velocity relative to the center of mass, gives the decomposition

K=\frac12 M V^2+\frac12\sum_i m_i u_i^2.

The first term is the kinetic energy of overall translation. The second is the kinetic energy of motion relative to the center of mass, including rotation and other internal motion. Because $\sum_i m_i\mathbf u_i=0$ , the cross terms vanish. For a given instantaneous state, the center-of-mass frame minimizes kinetic energy among frames related by a uniform change of velocity. (ocw.mit.edu)

For two particles, the relative-motion term becomes

K_{\mathrm{rel}} =\frac12\mu|\mathbf v_1-\mathbf v_2|^2, \qquad \mu=\frac{m_1m_2}{m_1+m_2}.

The quantity $\mu$ is the reduced mass. This form is particularly useful in collision calculations. In a perfectly inelastic collision between two particles that stick together, their final relative translational kinetic energy is zero, while the center-of-mass velocity remains unchanged if the net external impulse is negligible. (live.ocw.mit.edu)

Rotation about the center of mass

The center of mass is a natural reference point for rigid-body dynamics. The angular momentum about a fixed origin separates into the angular momentum associated with center-of-mass motion and the angular momentum relative to the center:

\mathbf L_O =\mathbf R\times\mathbf P+\mathbf L_{\mathrm{CM}},

where $\times$ denotes the cross product. This separates the body’s motion through space from its rotation about its own center of mass. (mitp-content-server.mit.edu)

The parallel-axis theorem relates the moment of inertia about an axis through the center of mass to that about a parallel axis:

I=I_{\mathrm{CM}}+Md^2,

where $d$ is the perpendicular separation of the axes. Thus, among axes with the same direction, the axis through the center of mass has the smallest moment of inertia. The theorem allows rotational properties of composite objects to be calculated from their components. (openstax.org)

Center of gravity and balance

The center of gravity concerns the resultant gravitational force and its torque, whereas the center of mass is defined by the mass distribution alone. In a uniform gravitational field they coincide: the body’s total weight can be represented as acting through its center of mass. In a field that varies appreciably across the body, the two need not coincide. (openstax.org)

For an object resting on a horizontal support under gravity, stability against tipping depends on the position of the center of gravity relative to the base of support. When its vertical projection passes beyond the supporting edge, gravity produces a torque that tends to overturn the object. A lower center of gravity or a wider base generally permits a greater tilt before this happens. These are statements about a specified support arrangement, not a universal guarantee of stability under arbitrary forces. (openstax.org)

Astronomy and the barycenter

In astronomy, the center of mass of a system of bodies is commonly called its barycenter. The term applies, for example, to the Solar System and to the Earth–Moon system. (ssd.jpl.nasa.gov)

In an isolated two-body orbital model, both bodies move about their common barycenter, rather than one remaining strictly stationary while the other orbits it. This is especially apparent in a binary star system. The barycenter lies closer to the more massive body and can lie inside it; its location does not have to coincide with either body’s geometric center. (science.nasa.gov)

Historical background

The mathematical treatment of balance predates the modern dynamical use of the center of mass. In On the Equilibrium of Planes, Archimedes studied the law of the lever and centers of gravity of plane figures. These problems concerned the conditions under which distributed weights balance. Newtonian mechanics gives the mass-weighted point a further role: it governs the overall translational motion of a system under external forces. (perseus.tufts.edu)

Limits and relativistic generalization

The simple equation $M\ddot{\mathbf R}=\mathbf F_{\mathrm{ext}}$ assumes a fixed collection of particles with constant masses. For an open system, such as a rocket considered without its expelled propellant, mass crossing the system boundary also transports momentum. Its equations of motion must account for that momentum flux; changing mass cannot simply be inserted into the constant-mass formula. (ocw.mit.edu)

In special relativity, rest-mass weighting alone does not capture the contributions of kinetic, interaction, and field energy. A related construction is the center of energy, expressed schematically for discrete energy contributions on a chosen simultaneous time slice as

\mathbf R_E=\frac{\sum_i E_i\mathbf r_i}{\sum_i E_i}.

A complete treatment includes all relevant energy, including electromagnetic field energy where present. For an isolated relativistic system, the center-of-energy motion is related to total momentum by

\frac{d\mathbf R_E}{dt} =\frac{c^2\mathbf P}{E}.

The relativistic center-of-momentum description therefore extends, rather than merely repeats, the Newtonian mass-weighted construction. (balitsky.com)

References

6 Center of Mass — University Physics Volume 1openstax.org
6 Moments and Centers of Mass — Calculus Volume 2openstax.org
6 Calculating Centers of Mass and Moments of Inertia — Calculus Volume 3openstax.org
223 IAP 2017 Lecture 5: Full Example and Conservation of Energyocw.mit.edu
3D Rigid Body Dynamics: Kinetic Energy, Instability, Equations of Motionocw.mit.edu
MIT8_01F16_DD_CMframe7_360plive.ocw.mit.edu
Structure and Interpretation of Classical Mechanics: Chapter 2mitp-content-server.mit.edu
5 Calculating Moments of Inertia — University Physics Volume 1openstax.org
1 Conditions for Static Equilibrium — University Physics Volume 1openstax.org
Ch. 12 Summary — University Physics Volume 1openstax.org
3 Stability — College Physics for AP Courses 2eopenstax.org
Barycenter — NASA Jet Propulsion Laboratoryssd.jpl.nasa.gov