Moments and Center of Mass

Definition of center of mass

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

'"`UNIQ--postMath-00000001-QINU`"'

Solving this equation for R yields the formula

'"`UNIQ--postMath-00000002-QINU`"'

where '"`UNIQ--postMath-00000003-QINU`"' is the total mass of all of the particles.

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

'"`UNIQ--postMath-00000004-QINU`"'

Solve this equation for the coordinates R to obtain

'"`UNIQ--postMath-00000005-QINU`"'

where M is the total mass in the volume.

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

Locating the center of mass

Plumb line method

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.

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.

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 compare with the required displacement and center of buoyancy of a ship, and ensure it would not capsize.

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.

Resources

• Center of mass, Wikipedia

Videos

Moments and Center of Mass of a Discrete Set of Objects by patrickJMT

Centroids / Centers of Mass - Part 1 of 2 by patrickJMT

Centroids / Centers of Mass - Part 2 of 2 by patrickJMT

Center of mass of the system, x-axis by Krista King

Center of mass of the system by Krista King

Centroids of Plane Regions by Krista King

Moments of the System by Krista King

Moment, Center of Mass, and Centroid by The Organic Chemistry Tutor

Center of Mass & Centroid Problems by The Organic Chemistry Tutor

Ex: Determine the Center of Mass of Three Point Masses in the Plane by James Sousa, Math is Power 4U

Ex: Find the Centroid of a Region Consisting of Three Rectangles by James Sousa, Math is Power 4U

Ex: Find the Centroid of a Triangular Region in the Plane by James Sousa, Math is Power 4U

Ex: Find the Centroid of a Bounded Region Involving Two Quadratic Functions by James Sousa, Math is Power 4U