mass distribution

(noun)

Describes the spatial distribution, and defines the center, of mass in an object.

Related Terms

Examples of mass distribution in the following topics:

  • Physics and Engeineering: Center of Mass

    • For a continuous mass distribution, the position of center of mass is given as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$ .
    • In physics, the center of mass (COM) of a mass or object in space is the unique point at which the weighted relative position of the distributed mass sums to zero.
    • In this case, the distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates.
    • If the mass distribution is continuous with respect to the density, $\rho (r)$, within a volume, $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, $\mathbf{R}$, is zero, that is:
    • If a continuous mass distribution has uniform density, which means $\rho$ is constant, then the center of mass is the same as the centroid of the volume.

  • Locating the Center of Mass

    • The center of mass is a statement of spatial arrangement of mass (i.e. distribution of mass within the system).
    • The position of COM is given a mathematical formulation which involves distribution of mass in space:
    • If the mass distribution is continuous with the density ρ(r) within a volume V, the position of COM is given as
    • If a continuous mass distribution has uniform density, which means ρ is constant, then the center of mass is the same as the center of the volume.
    • Identify the center of mass for an object with continuous mass distribution

  • Weight of the Earth

    • where $F$ is the force between the masses, $G$ is the gravitational constant, $m_1$ is the first mass, $m_2$ is the second mass and $r$ is the distance between the centers of the masses.
    • In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.
    • For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force.
    • The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance $r_0$ from the center of the mass distribution:
    • The portion of the mass that is located at radii $rmass enclosed within a sphere of radius $r_0$ was concentrated at the center of the mass distribution (as noted above).

  • Center of Mass and Inertia

    • 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 the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
    • If the mass distribution is continuous with the density $\rho (r)$ within a volume $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass $\mathbf{R}$ is zero; that is:
    • where $M$ is the total mass in the volume.
    • Use multiple integrals to find the center of mass of a distribution of mass

  • The Physical Pendulum

    • Gravity acts through the center of mass of the rigid body.
    • However, it is not independent of the mass distribution of the rigid body.
    • A change in shape, size, or mass distribution will change the moment of inertia.
    • An example showing how forces act through center of mass.
    • A rigid rod with uniform mass distribution hangs from a pivot point.

  • Applications of Multiple Integrals

    • The gravitational potential associated with a mass distribution given by a mass measure $dm$ on three-dimensional Euclidean space $R^3$ is:
    • If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
    • In the following example, the electric field produced by a distribution of charges given by the volume charge density $\rho (\vec r)$ is obtained by a triple integral of a vector function:
    • This can also be written as an integral with respect to a signed measure representing the charge distribution.
    • Points $\mathbf{x}$ and $\mathbf{r}$, with $\mathbf{r}$ contained in the distributed mass (gray) and differential mass $dm(\mathbf{r})$  located at the point $\mathbf{r}$.

  • Gravitational Attraction of Spherical Bodies: A Uniform Sphere

    • When considering the gravitational force exerted on an object at a point inside or outside a uniform spherically symmetric object of radius $R$, there are two simple and distinct situations that must be examined: the case of a hollow spherical shell, and that of a solid sphere with uniformly distributed mass.
    • Only the mass of the sphere within the desired radius $M_{mass of the sphere inside $d$) is relevant, and can be considered as a point mass at the center of the sphere.
    • That is, the sphere's mass is uniformly distributed.)
    • As in the case of hollow spherical shells, the net gravitational force that a solid sphere of uniformly distributed mass $M$ exerts on a body outside of it, is the vector sum of the gravitational forces acted by each shell of the sphere on the outside object.
    • More generally, this result is true even if the mass $M$ is not uniformly distributed, but its density varies radially (as is the case for planets).

  • Moment of Inertia

    • The moment of inertia is a property of the distribution of mass in space that measures mass's resistance to rotational acceleration about one or more axes.
    • The moment of inertia I of an object can be defined as the sum of mr2 for all the point masses of which it is composed, where m is the mass and r is the distance of the mass from the center of mass.
    • Assuming that the hoop material is uniform, the hoop's moment of inertia can be found by summing up all the mass of the hoop and multiplying by the distance of that mass from the center of mass.
    • All of the mass m is at a distance r from the center.
    • The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates.

  • Distribution of Molecular Speeds and Collision Frequency

    • In the above formula, R is the gas constant, T is absolute temperature, and Mm is the molar mass of the gas particles in kg/mol.
    • In theory, this energy can be distributed among the gaseous particles in many ways, and the distribution constantly changes as the particles collide with each other and with their boundaries.
    • Velocity distributions are dependent on the temperature and mass of the particles.
    • Explore the role of molecular mass on the rate of diffusion.
    • Select the mass of the molecules behind the barrier.

  • Characteristics of Mass Spectra

    • Most of the ions formed in a mass spectrometer have a single charge, so the m/z value is equivalent to mass itself.
    • Modern mass spectrometers easily distinguish (resolve) ions differing by only a single atomic mass unit (amu), and thus provide completely accurate values for the molecular mass of a compound.
    • The highest-mass ion in a spectrum is normally considered to be the molecular ion, and lower-mass ions are fragments from the molecular ion, assuming the sample is a single pure compound.
    • The molecules of these compounds are similar in size, CO2 and C3H8 both have a nominal mass of 44 amu, and C3H6 has a mass of 42 amu.
    • Both distributions are observed, but the larger ethyl cation (m/z=29) is the most abundant, possibly because its size affords greater charge dispersal.