Center of mass
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In physics, the center 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 presence of a uniform gravitational field, the center of mass acts as a center of gravity; the net gravitational torque on a system is equal to the torque resulting in the system's weight applied at the CM.
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!
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Definition
The center of mass R of a system of particles is defined as the average of their positions ri, weighted by their masses mi:
- <math>\mathbf{R} = \frac 1M \sum m_i \mathbf{r}_i</math>
where M is the total mass of the system, equal to the sum of the particle masses.
For a continuous distribution with mass density ρ(r), the sum becomes an integral:
- <math>\mathbf R =\frac 1M \int \mathbf{r} \; dm = \frac 1M \int\rho(\mathbf{r})\, \mathbf{r} \ dV.</math>
If an object has uniform density then its center of mass is the same as the centroid of its shape.
Examples
- The CM of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The CM is closer to the more massive object; for details, see #Barycenter below.
- The CM of a ring is at the center of the ring (in the air).
- The CM of a solid triangle lies on all three medians and therefore at the centroid, which is also the average of the three vertices.
- The CM of a rectangle is at the intersection of the two diagonals. This principle is used in the example given below [Locating center of mass]
- 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 CM will be a fixed point of that symmetry.
Locating center of mass
Here is an interesting way of determining the CM of an 'L' shaped 2-D object as given in fig 1:
1) Divide the shape into two rectangles, as shown in fig 2. Find the CMs of these two rectangles by drawing the diagonals. Draw a line joining the CMs. The CM of the 'L' shape must lie on this line AB.
2) Divide the shape into two other rectangles, as shown in fig 3. Find the CMs of these two rectangles by drawing the diagonals. Draw a line joining the CMs. The CM of the 'L' shape must lie on this line CD.
3) As the CG of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. The point 'O' might not lie inside the L-shaped object.
(see also Center of Gravity for an experimental method determining Center of Gravity in 2-D objects, which in most cases will be equivalent to Center of Mass)
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
- <math>\mathbf{p}=M\mathbf{v}_\mathrm{cm}</math>
Where M indicates the total mass, and vcm 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
- <math>\mathbf{F} = M\mathbf{a}_\mathrm{cm}</math>
Where F indicates the sum of all external forces on the system, and acm indicates the acceleration of the center of mass.
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 <math>M</math>:
- <math>\mathbf{L}_\mathrm{sys} = \mathbf{L}_\mathrm{cm} + \mathbf{L}_\mathrm{around\,cm}</math>
Rotation and centers of gravity
In physics, the center of gravity (CoG) of a rigid body is the averaged location of its weight. In a uniform gravitational field it coincides with the object's center of mass, but in a nonuniform gravitational field it can be located elsewhere.
For most practical Earth-dwelling purposes, Centre of mass is equivalent to Centre of gravity (with subtle differences). If you hang an object from a string, the object's CM (/CG) will be directly below the string, and in line with it.
Image:CoG stable.jpg 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 is always tends to come below the point used as the support such that the object will never topple. Many educational toys rely on this fact. Fig 1 is such an example. C is the CM of the toy and P is the point where it is supported.
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.
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.
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, 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:
- <math>r_1 = r_{\rm tot} {m_2 \over m_1 + m_2}</math>
where :
- r1 is the distance from body 1 to the barycenter
- rtot 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:
- <math>\frac{0 \times 1 + 384400\mbox{ km} \times 0.0123}{1 + 0.0123} = 4671\mbox{ km}</math>
- 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 11 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.
Image:Orbit1.gif Image:Orbit2.gif Image:Orbit3.gif Image:Orbit4.gif Image:Orbit5.gif Template:Clear
See also
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.cs:Těžiště
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