Center of mass

In

linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion

.

In the case of a single

planets of the Solar System

, the center of mass may not correspond to the position of any individual member of the system.

The center of mass is a useful reference point for calculations in

inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system

.

History

The concept of center of gravity or weight was studied extensively by the ancient Greek mathematician, physicist, 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 his work On Floating Bodies, Archimedes 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.^[1]

Other ancient mathematicians who contributed to the theory of the center of mass include

Jean-Charles de la Faille, Paul Guldin,^[6] John Wallis, Christiaan Huygens,^[7] Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.^[8]

Euler's first law.^[9]

Definition

The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space.

A system of particles

In the case of a system of particles $P i, i = 1, ..., n$ , each with mass $m i$ that are located in space with coordinates $r i, i = 1, ..., n$ , the coordinates R of the center of mass satisfy the condition

\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .

Solving this equation for R yields the formula

\mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.

A continuous volume

If the mass distribution is continuous with the density ρ(r) within a solid Q, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass R over the volume V is zero, that is

\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0.

Solve this equation for the coordinates R to obtain

\mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,

Where M is the total mass in the volume.

If a continuous mass distribution has uniform density, which means that ρ is constant, then the center of mass is the same as the centroid of the volume.^[10]

Barycentric coordinates

The coordinates R of the center of mass of a two-particle system, P₁ and P₂, with masses m₁ and m₂ is given by

\mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.

Let the percentage of the total mass divided between these two particles vary from 100% P₁ and 0% P₂ through 50% P₁ and 50% P₂ to 0% P₁ and 100% P₂, then the center of mass R moves along the line from P₁ to P₂. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, and are termed barycentric coordinates. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

Systems with periodic boundary conditions

For particles in a system with periodic boundary conditions two particles can be neighbours even though they are on opposite sides of the system. This occurs often in molecular dynamics simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, x and y and/or z, as if it were on a circle instead of a line.^[11] The calculation takes every particle's x coordinate and maps it to an angle,

\theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi

where x_max is the system size in the x direction and

x_{i}\in [0,x_{\max })

. From this angle, two new points

(\xi _{i},\zeta _{i})

can be generated, which can be weighted by the mass of the particle

x_{i}

for the center of mass or given a value of 1 for the geometric center:

{\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}

In the $(\xi ,\zeta )$ plane, these coordinates lie on a circle of radius 1. From the collection of $\xi _{i}$ and $\zeta _{i}$ values from all the particles, the averages ${\overline {\xi }}$ and ${\overline {\zeta }}$ are calculated.

{\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\end{aligned}}

where

M

is the sum of the masses of all of the particles.

These values are mapped back into a new angle, ${\overline {\theta }}$ , from which the x coordinate of the center of mass can be obtained:

{\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\end{aligned}}

The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using cluster analysis to "unfold" a cluster straddling the periodic boundaries. If both average values are zero, $\left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)$ , then ${\overline {\theta }}$ is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their x coordinates are mathematically identical in a periodic system.

Center of gravity

A body's center of gravity is the point around which the resultant torque due to gravity forces vanishes. Where a gravity field can be considered to be uniform, the mass-center and the center-of-gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer-to and further-from the planet (stronger and weaker gravity respectively) can lead to a torque that will tend to align the satellite such that its long axis is vertical. In such a case, it is important to make the distinction between the center-of-gravity and the mass-center. Any horizontal offset between the two will result in an applied torque.

The mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center-of-gravity will always be located somewhat closer to the main attractive body as compared to the mass-center, and thus will change its position in the body of interest as its orientation is changed.

In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass-center as the center-of-gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably.

In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the resultant of the gravity forces on a continuous body. Consider a body Q of volume V with density ρ(r) at each point r in the volume. In a parallel gravity field the force f at each point r is given by,

\mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,

where dm is the mass at the point r, g is the acceleration of gravity, and

{\textstyle \mathbf {\hat {k}} }

is a unit vector defining the vertical direction.

Choose a reference point R in the volume and compute the resultant force and torque at this point,

\mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}} ,

and

\mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).

If the reference point R is chosen so that it is the center of mass, then

\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,

which means the resultant torque T = 0. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass.

Linear and angular momentum

The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P_i, i = 1, ..., n of masses m_i be located at the coordinates r_i with velocities v_i. Select a reference point R and compute the relative position and velocity vectors,

\mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v} .

The total linear momentum and angular momentum of the system are

\mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} ,

and

\mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v}

If R is chosen as the center of mass these equations simplify to

\mathbf {p} =m\mathbf {v} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v}

where m is the total mass of all the particles, p is the linear momentum, and L is the angular momentum.

The

Newton's Third Law.^[12]

Determination

The experimental determination of a body's centre of mass makes use of gravity forces on the body and is based on the fact that the centre of mass is the same as the centre of gravity in the parallel gravity field near the earth's surface.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.^[13]

In two dimensions

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.^[14]

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.[15] This method can even work for objects with holes, which can be accounted for as negative masses.^[16]

A direct development of the

center of buoyancy of a ship, and ensure it would not capsize.^[17]^[18]

In three dimensions

An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, F₁, F₂, and F₃ that resist the weight of the object, $\mathbf {W} =-W\mathbf {\hat {k}}$ ( $\mathbf {\hat {k}}$ is the unit vector in the vertical direction). Let r₁, r₂, and r₃ be the position coordinates of the support points, then the coordinates R of the center of mass satisfy the condition that the resultant torque is zero,

\mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,

or

\mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.

This equation yields the coordinates of the center of mass R* in the horizontal plane as,

\mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).

The center of mass lies on the vertical line L, given by

\mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object. The center of mass will be the intersection of the two lines L₁ and L₂ obtained from the two experiments.

Applications

Engineering designs

Automotive applications

Engineers try to design a

handle

better, which is to say, maintain traction while executing relatively sharp turns.

The characteristic low profile of the U.S. military Humvee was designed in part to allow it to tilt farther than taller vehicles without rolling over, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the horizontal.

Aeronautics

The center of mass is an important point on an

stalled condition.^[20]

For

hover, the center of mass is always directly below the rotorhead. In forward flight, the center of mass will move forward 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.^[21]

Astronomy

Retrieved from "https://en.wikipedia.org/w/index.php?title=Center_of_mass&oldid=1200545034"

This page is based on the copyrighted Wikipedia article: Center of mass. Articles is available under the CC BY-SA 3.0 license; additional terms may apply.Privacy Policy

[1]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[16]

[17]

[18]

[20]

[21]

Rigging and safety

Body motion

Optimization

See also

Notes

References

External links