Center of mass

In physics, the center of mass (or centre of mass) of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system. In the case of a rigid body, its center of mass is rigidly fixed to the object. In the context of a uniform gravitational field, the center of mass is sometimes called the center of gravity, since the net gravitational torque on a system is equal to the torque resulting in the system's weight applied at the center of mass.

The center of mass of a body does not always coincide with its intuitive geometric center, and one can exploit this freedom. Engineers try hard to make a sport car as light as possible, and then add weight on the bottom; this way, the center of mass is nearer to the street, and the car handles better. When high jumpers perform a "Fosbury Flop," they bend their body in such a way that it is possible for the jumper to clear the bar while his or her center of mass does not.

Definition

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

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

where

M

is the total mass of the system, equal to the sum of the particle masses.

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 #Barycenter below.
The center of mass of a ring is at the center of the ring (in the air).
The center of mass of a 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 rectangle is at the intersection of the two diagonals.
In a spherically symmetric body, the center of mass is at the center. This approximately applies to the Earth: the density varies considerably, but it mainly depends on depth and less on the other two coordinates.
More generally, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

History

The concept of center of gravity was first introduced by the ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse. 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 gravity. In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of gravity as low as possible. He developed mathematical techniques for finding the centers of gravity of objects of uniform density of various well-defined shapes, in particular a triangle, a hemisphere, and a frustum of a circular paraboloid.

Locating the center of mass of an arbitrary 2D physical shape

This method is useful when you wish to find the center of gravity of a complex planar object with unknown dimensions.

Step 1: An arbitrary 2D shape.

Step 2: Suspend the shape from a location near an edge. Drop a plumb line and mark on the object.

Step 3: Suspend the shape from another location not too close to the first. Drop a plumb line again and mark. The intersection of the two lines is the center of gravity.

Locating the center of mass of a composite shape

This method is useful when you wish to find the center of gravity of an object which is easily divided into elementary shapes. See: List of centroids. We will only be finding the center of mass in the x direction here. The same procedure should be followed to locate the center of mass in the y direction, and is left as an exercise for the reader.

The shape. It is easily divided into a square, triangle, and circle. Note that the circle will have negative area.

From the List of centroids, we note the coordinates of the individual centroids.

From equation 1 above: $\frac{-3 \times \pi \times 2.5^2 + 5 \times 10^2 + 13.33 \times \frac{10^2}{2}}{ -\pi \times 2.5^2 + 10^2 + \frac{10^2}{2}} \approx 8.5$ units.

The centre of mass of this figure is at a distance of 8.5 units from the left corner of the figure.

Motion

The following equations of motion assume that there is a system of particles governed by internal and external forces. An internal force is a force caused by the interaction of the particles within the system. An external force is a force that originates from outside the system, and acts on one or more particles within the system. The external force need not be due to a uniform field.

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 the weak form of 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 the famous 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.

Rotation and centers of gravity

The center of mass is often called the center of gravity because for at least two purposes, any uniform (constant) gravitational field g acts on a system as if the mass were concentrated at the CM:

The gravitational potential energy of a system equals the potential energy of a point mass M at R.
The gravitational torque on a system equals the torque of a force Mg acting at R:
$M\mathbf{g}\times \mathbf{R}=\sum_im_i\mathbf{g}\times \mathbf{r}_i.$

When the CM of an object is directly under or over the base (or support), the object is said to be in a state of stable equilibrium. It is possible to construct an object whose CM always tends to come below the point used as the support such that the object will never topple.

If the gravitational field acting on a body is not uniform, then the center of mass does not necessarily exhibit these convenient properties concerning gravity. As the situation is put in Feynman's influential textbook The Feynman Lectures on Physics:

"The center of mass is sometimes called the center of gravity, for the reason that, in many cases, gravity may be considered uniform. ...In case the object is so large that the nonparallelism of the gravitational forces is significant, then the center where one must apply the balancing force is not simple to describe, and it departs slightly from the center of mass. That is why one must distinguish between the center of mass and the center of gravity."

Later authors are often less careful, stating that when gravity is not uniform, "the center of gravity" departs from the CM. This usage seems to imply a well-defined "center of gravity" concept for non-uniform fields, but there is no such thing. Even when considering tidal forces on planets, it is sufficient to use centers of mass to find the overall motion. In practice, for non-uniform fields, one simply does not speak of a "center of gravity".

CM frame

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}

Engineering

Aeronautical significance

The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is safe to fly, it is critical that the center of gravity fall within specified limits. This range varies by aircraft, but as a rule of thumb it is centered about a point one quarter of the way from the wing leading edge to the wing trailing edge (the quarter chord point). 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 moment arm of the elevator is reduced, which makes it more difficult to recover from a stalled condition. The aircraft will be more maneuverable, but also less stable, and possibly so unstable that it is impossible to fly.

Barycenter

When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planet, or a planet orbits around a star, both of them are actually orbiting around their center of mass, called the barycenter (or barycentre), see two-body problem.

The barycenter (from the Greek βαρύκεντρον) is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. It is an important concept in the fields of astronomy, astrophysics, and the like.

In the case where one of the two objects is much larger and more massive than the other, the barycenter will be located within the larger object. Rather than appearing to orbit it will simply be seen to "wobble" slightly. This is the case for the Moon and Earth, where the barycenter is located on average 4,671 km from Earth's center, well within the planet's radius of 6,378 km. When the two bodies are of similar masses (or at least the mass ratio is less extreme), however, the barycenter will be located outside of either of them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, Jupiter and the Sun, and many binary asteroids and binary stars.

The distance from the center of a body (thought of as a point-mass) to the barycenter in a simple two-body case can be calculated as follows:

r_1 = r_{\rm tot} {m_2 \over m_1 + m_2}

where :

r₁ is the distance from body 1 to the barycenter

r_tot is the distance between the two bodies

m₁ and m₂ are the masses of the two bodies.

Some examples:

Earth-Moon system: the Moon's mass is 0.0123 that of Earth. Put Earth in position 0, mass 1 (here we use an arbitrary mass unit. It does not matter, provided that we use the same unit for the Moon). The Moon is at an average distance of 384400 km from the Earth. Then the center of mass is at:

\frac{0 \times 1 + 384400\mbox{ km} \times 0.0123}{1 + 0.0123} = 4671\mbox{ km}

from the Earth's center. Thus, as opposed to the Earth standing "still" and the Moon moving, both of them move around a point about 1700 km below the Earth's surface.

Sun-Earth system: put Sun in position 0, mass=333,000 times the Earth. Earth in position 150,000,000 km, mass=1. Center of mass is 450 km from the Sun center. Here, the large mass difference between the two bodies makes the center of mass lie almost at the center of the Sun.

Sun-Jupiter system: put Sun in position 0, mass = 333,000 Earths. Jupiter in position 778,000,000 km, mass=318 Earths. Center of mass is 742,000 km from the Sun center, 46,000 km outside its surface. As Jupiter does its 12 year orbit, the Sun does a 1.5 million km orbit around the center of mass.

To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the solar system.

Note that the distance from the Sun's center to the center of mass of a two-body system consisting of the Sun and another celestial body, hence the size of the Sun's orbit around this center of mass, is approximately proportional to the product of the mass of that other body, and the distance between the two, even though gravity decreases with distance. That orbit is largest with Jupiter, its large mass more than compensates its smaller distance to the Sun than several other planets. If all the planets would align on the same side of the Sun, the combined center of mass would lie about 500,000 km outside the Sun surface.

Animations

Images are representative, not simulated.

From: Lourence Joseph Navarro

References

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.

Classical mechanics | Mass | Means

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