Center of mass

This article is about physical objects. For the military concept, see Center of gravity (military). For barycenters in geometry, see centroid.

A child's toy uses the principles of center of mass to keep balance on a finger

In physics, the center of mass or barycenter of a body is a point in space where, for the purpose of various calculations, the entire mass of a body is concentrated. The center of gravity is a related point where the gravitational weight of a body acts as if it were concentrated. In a uniform gravitational field this point is found at the center of mass, and In common usage, the two points are considered the same. In a non-uniform field, the center of mass no longer serves as the exact center of gravity, so physicists often distinguish the center of gravity as a separate concept.

The center of mass of a body may be defined as the average location of the mass distribution. In the case of a rigid body, the center of mass is fixed in relation to the body, and it may or may not coincide with the geometric center. In the case of a loose distribution of masses, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual mass.

The mass center often obeys simple equations of motion, and it is a convenient reference point for many other calculations in mechanics, such as angular momentum and moment of inertia. In many applications, such as orbital mechanics, objects can be replaced by point masses located at their mass centers for the purposes of analysis. The center of mass frame is an inertial frame in which the center of mass of a system is at rest at the origin of the coordinate system.

In a non-uniform field, gravitational effects such as potential energy, force, and torque can no longer be calculated using the center of mass alone. In particular, a non-uniform gravitational field can produce a torque on an object, causing it to rotate. The center of gravity seeks to explain this effect. Formally, a center of gravity is an application point of the resultant gravitational force on the body. Such a point may not exist, and if it exists, it is not unique. One can further define a unique center of gravity by approximating the field as either parallel or spherically symmetric. Therefore, the concept of a center of gravity as distinct from the center of mass is rarely used in applications, even in celestial mechanics, where non-uniform fields are important. Since the center of gravity depends on the external field, its motion is harder to determine than the motion of the center of mass. The common method to deal with gravitational torques is a field theory.

Definition

The center of mass $\mathbf{R}$ of a system of particles of total mass M is defined as the average of their positions, $\mathbf{r}_i$, weighted by their masses, m_i:

$\mathbf{R} = \frac{1}{M} \sum m_i \mathbf{r}_i.$

For a continuous distribution with mass density $\rho(\mathbf{r})$, the sum becomes an integral:

$\mathbf R =\frac 1M \int \mathbf{r} \; dm = \frac 1M \int\rho(\mathbf{r})\, \mathbf{r} \ dV.$

If an object has uniform density then its center of mass is the same as the centroid of its shape.

Examples

• The center of mass of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The center of mass is closer to the more massive object; for details, see below.
• The center of mass of a uniform ring is at the center of the ring; outside the material that makes up the ring.
• The center of mass of a uniform solid triangle lies on all three medians and therefore at the centroid, which is also the average of the three vertices.
• The center of mass of a uniform rectangle is at the intersection of the two diagonals.
• In a spherically symmetric body, the center of mass is at the geometric center. This approximately applies to the Earth: the density varies considerably, but it mainly depends on depth and less on the latitude and longitude coordinates.

More generally, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

Properties

Momentum

Further information: Derivation of the center of mass, Center of mass frame

For any system with no external forces, the center of mass moves with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that satisfy Newton's Third Law.

The total momentum for any system of particles is given by

$\mathbf{p}=M\mathbf{v}_\mathrm{cm},$

where M indicates the total mass, and v_cm is the velocity of the center of mass. This velocity can be computed by taking the time derivative of the position of the center of mass. An analogue to Newton's Second Law is

$\mathbf{F} = M\mathbf{a}_\mathrm{cm},$

where F indicates the sum of all external forces on the system, and a_cm indicates the acceleration of the center of mass. It is this principle that gives precise expression to the intuitive notion that the system as a whole behaves like a mass of M placed at R.

The angular momentum vector for a system is equal to the angular momentum of all the particles around the center of mass, plus the angular momentum of the center of mass, as if it were a single particle of mass M:

$\mathbf{L}_\mathrm{sys} = \mathbf{L}_\mathrm{cm} + \mathbf{L}_\mathrm{around\,cm}.$

This is a corollary of the parallel axis theorem.

Gravity

Diagram of an educational toy that balances on a point: the CM (C) settles below its support (P).

The suspending chair trick makes use of the human body's surprisingly high center of mass.

The center of mass is often called the center of gravity because any uniform gravitational field g acts on a system as if the mass M of the system were concentrated at the center of mass R. Specifically, the gravitational potential energy is equal to the potential energy of a point mass M at R, and the gravitational torque is equal to the torque of a force Mg acting at R.

History

The concept of a center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point — their center of mass. In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.

Later mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.

Newton's second law is reformulated with respect to the center of mass in Euler's first law.

Locating the center of mass

Main article: Locating the center of mass

Plumb line method

An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass.

The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers. This method can even work for objects with holes, which can be accounted for as negative masses.

A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to ensure the ship would not capsize.

Applications

Engineers try to design a sports car's center of mass as low as possible to make the car handle better. When high jumpers perform a "Fosbury Flop", they bend their body in such a way that it clears the bar while its center of mass does not.

Aeronautics

Main article: Center of gravity of an aircraft

The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within specified limits. If the center of mass is ahead of the forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing. If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly. The moment arm of the elevator will also be reduced, which makes it more difficult to recover from a stalled condition.

For helicopters in hover, the center of mass is always directly below the rotorhead. In forward flight, the center of mass will move aft to balance the negative pitch torque produced by applying cyclic control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.

Astronomy

Two bodies orbiting a barycenter internal to one body

Main article: Barycentric coordinates (astronomy)

The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies outside the center of the primary (the larger body). For example, the moon does not orbit the exact center of the Earth, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km (1062 miles) below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun.

Centers of gravity

If the gravitational field acting on a body is not uniform, then a non-uniform gravitational field can produce a torque on an object about its center of mass, causing it to rotate. The center of gravity seeks to model the gravitational torque as a resultant force at a point. Such a point may not exist, and if it exists, it is not unique. When a unique center of gravity can be defined, its location depends on the external field, so its motion is harder to determine than the motion of the center of mass; this problem limits its usefulness in applications.

Textbooks such as the The Feynman Lectures on Physics characterize the center of gravity as a point about which there is no torque. In other words, the center of gravity is a point of application for the resultant force. Under this formulation, the center of gravity r_cg is defined as a point that satisfies the equation

$\mathbf{r}_\mathrm{cg} \times \mathbf{F} = \boldsymbol{\tau},$

where F and τ are the total force and torque on the body due to gravity.

One complication concerning r_cg is that its defining equation is not generally solvable. If F and τ are not orthogonal, then there is no solution; the force of gravity does not have a resultant and cannot be replaced by a single force at any point. There are some important special cases where F and τ are guaranteed to be orthogonal, such as if all forces lie in a single plane or are aligned with a single point.

If the equation is solvable, there is another complication: its solutions are not unique. Instead, there are infinitely many solutions; the set of all solutions is known as the line of action of the force. This line is parallel to the weight F. In general, there is no way to choose a particular point as the unique center of gravity. A single point may still be chosen in some special cases, such as if the gravitational field is parallel or spherically symmetric. These cases are considered below.

Parallel fields

Some of the inhomogeneity in a gravitational field may be modeled by a variable but parallel field: g(r) = g(r)n, where n is some constant unit vector. Although a non-uniform gravitational field cannot be exactly parallel, this approximation can be valid if the body is sufficiently small. The center of gravity may then be defined as a certain weighted average of the locations of the particles composing the body. Whereas the center of mass averages over the mass of each particle, the center of gravity averages over the weight of each particle:

$\mathbf{r}_\mathrm{cg} = \frac{1}{W} \sum_i w_i \mathbf{r}_i,$

where w_i is the (scalar) weight of the ith particle and W is the (scalar) total weight of all the particles. This equation always has a unique solution, and in the parallel-field approximation, it is compatible with the torque requirement.

A common illustration concerns the Moon in the field of the Earth. Using the weighted-average definition, the Moon has a center of gravity that is lower (closer to the Earth) than its center of mass, because its lower portion is more strongly influenced by the Earth's gravity.

Spherically symmetric fields

If the external gravitational field is spherically symmetric, then it is equivalent to the field of a point mass M at the center of symmetry r. In this case, the center of gravity can be defined as the point at which the total force on the body is given by Newton's Law:

$\frac {GmM (\mathbf{r}_\mathrm{cg} - \mathbf{r})} {|\mathbf{r}_\mathrm{cg} - \mathbf{r}|^3} = \mathbf{F},$

where G is the gravitational constant and m is the mass of the body. As long as the total force is nonzero, this equation has a unique solution, and it satisfies the torque requirement. A convenient feature of this definition is that if the body is itself spherically symmetric, then r_cg lies at its center of mass. In general, as the distance between r and the body increases, the center of gravity approaches the center of mass.

Another way to view this definition is to consider the gravitational field of the body; then r_cg is the apparent source of gravitational attraction for an observer located at r. For this reason, r_cg is sometimes referred to as the center of gravity of M relative to the point r.

Usage

The centers of gravity defined above are not fixed points on the body; rather, they change as the position and orientation of the body changes. This characteristic makes the center of gravity difficult to work with, so the concept has little practical use.

When it is necessary to consider a gravitational torque, it is easier to represent gravity as a force acting at the center of mass, plus an orientation-dependent couple. The latter is best approached by treating the gravitational potential as a field.

See also

• Center of percussion
• Center of pressure
• Mass point geometry
• Metacentric height
• Roll center
• Weight distribution

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External links

• Motion of the Center of Mass shows that the motion of the center of mass of an object in free fall is the same as the motion of a point object.
• The Solar System's barycenter Simulations showing the effect each planet contributes to the Solar System's barycenter