center of mass

(noun)

The center of mass (COM) 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.

Related Terms

Examples of center of mass in the following topics:

  • 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).
    • 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.
    • The intersection of the two lines is the center of mass .
    • The intersection of the two lines is the center of mass.

  • Center of Mass and Inertia

    • The center of mass for a rigid body can be expressed as a triple integral.
    • In this atom, we will see how center of mass can be calculated using multiple integrals.
    • 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.
    • Two bodies orbiting around the center of mass inside one body
    • Use multiple integrals to find the center of mass of a distribution of mass

  • Center of Gravity

    • The center of gravity is read mathematically as: 'the position of the center of mass and weighted average of the position of the particles'.
    • Three-dimensional bodies have a property called the center of mass, or center of gravity.
    • The center of mass does not actually carry all the mass, despite appearances.
    • Specifically: 'the total mass x the position of the center of mass= ∑ the mass of the individual particle x the position of the particle. ' The center of mass is a geometric point in three-dimensional volume.
    • Describe how the center of mass of an oddly shaped object is found

  • 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 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.
    • The two objects are rotating around their center of mass.

  • Center of Mass and Translational Motion

    • The COM (center of mass) of a system of particles is a geometric point that assumes all the mass and external force(s) during motion.
    • This point is known as center of mass, abbreviated COM (the mathematical definition of COM will be introduced in the next Atom on "Locating the Center of Mass").
    • The center of mass appears to carry the whole mass of the body.
    • At the center of mass, all external forces appear to apply.
    • We can describe general motion of an object (with mass m) as follows:

  • The Physical Pendulum

    • The period of a physical pendulum depends upon its moment of inertia about its pivot point and the distance from its center of mass.
    • Gravity acts through the center of mass of the rigid body.
    • where h is the distance from the center of mass to the pivot point and θ is the angle from the vertical.
    • Clearly, the center of mass is at a distance L/2 from the point of suspension:
    • An example showing how forces act through center of mass.

  • Motion of the Center of Mass

    • We can describe the translational motion of a rigid body as if it is a point particle with the total mass located at the COM—center of mass.
    • We can describe the translational motion of a rigid body as if it is a point particle with the total mass located at the center of mass (COM).
    • In this Atom. we will prove that the total mass (M) times the acceleration of the COM (aCOM), indeed, equals the sum of external forces.
    • You can see that the Newton's 2nd law applies as if we are describing the motion of a point particle (with mass M) under the influence of the external force.
    • Derive the center of mass for the translational motion of a rigid body

  • Gravitational Attraction of Spherical Bodies: A Uniform Sphere

    • The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center.
    • So when finding the force of gravity exerted on a ball of 10 kg, the distance measured from the ball is taken from the ball's center of mass to the earth's center of 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.
    • which shows that mass $m$ feels a force that is linearly proportional to its distance, $d$, from the sphere's center of mass.
    • The resulting net gravitational force acts as if mass $M$ is concentrated on a point at the center of the sphere, which is the center of mass, or COM (Statement 1 of Shell Theorem).

  • The Law of Universal Gravitation

    • Objects with mass feel an attractive force that is proportional to their masses and inversely proportional to the square of the distance.
    • The Law of Universal Gravitation states that every point mass attracts every other point mass in the universe by a force pointing in a straight line between the centers-of-mass of both points, and this force is proportional to the masses of the objects and inversely proportional to their separation This attractive force always points inward, from one point to the other.
    • For these cases the mass of each object can be represented as a point mass located at its center-of-mass.
    • Because of the magnitude of $G$, gravitational force is very small unless large masses are involved.
    • The force is proportional to the masses and inversely proportional to the square of the distance.

  • 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.
    • 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 $rof the mass enclosed within a sphere of radius $r_0$ was concentrated at the center of the mass distribution (as noted above).
    • The portion of the mass that is located at radii $r>r_0$ exerts no net gravitational force at the distance $r_0$ from the center.