point mass

(noun)

A theoretical point with mass assigned to it.

Related Terms

Examples of point mass in the following topics:

  • Weight of the Earth

    • When the bodies have spatial extent, gravitational force is calculated by summing the contributions of point masses which constitute them.
    • Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.
    • In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points.
    • If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies.
    • 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 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.
    • Two big objects can be considered as point-like masses, if the distance between them is very large compared to their sizes or if they are spherically symmetric.
    • For these cases the mass of each object can be represented as a point mass located at its center-of-mass.
    • All masses are attracted to each other.

  • Gravitational Attraction of Spherical Bodies: A Uniform Sphere

    • The Law of Universal Gravitation states that the gravitational force between two points of mass is proportional to the magnitudes of their masses and the inverse-square of their separation, $d$:
    • However, most objects are not point particles.
    • 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.
    • So, the gravitational force acting upon point mass $m$ is:
    • 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).

  • 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.
    • We describe the translational motion of a rigid body as if it is a point particle with mass m located at COM.
    • By introducing the concept of COM, the translational motion becomes that of a point particle with mass m.

  • 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).
    • 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.
    • The COM will orbit around the Sun as if it is a point particle.
    • Derive the center of mass for the translational motion of a rigid body

  • 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.
    • 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:
    • However, we need to evaluate the moment of inertia about the pivot point, not the center of mass, so we apply the parallel axis theorem:
    • A rigid rod with uniform mass distribution hangs from a pivot point.

  • 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).
    • 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.
    • In three dimensions: By supporting an object at three points and measuring the forces that resist the weight of the object, COM of the three-dimensional coordinates of the center of mass can be determined.
    • Suspend the object from two locations and to drop plumb lines from the suspension points.

  • Center of Gravity

    • This is because the center of mass is at the point where people hold it up with their fingers.
    • The position of this force causes the object to act as a single point of force from the point.
    • The different parts of the body have different motions. shows the motion of a stick in the air: it seems to rotate around a single point.
    • 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.

  • Center of Mass of the Human Body

    • The center of mass (COM) is an important physical concept—it is the point about which objects rotate.
    • The center of mass (COM) is an important physical concept.
    • It is the point on an object at which the weighted relative position of the distributed mass sums to zero—the point about which objects rotate.
    • The center of mass of the human body depends on the gender and the position of the limbs.
    • where M is mass of the subject.

  • Inelastic Collisions in Multiple Dimensions

    • At this point we will expand our discussion of inelastic collisions in one dimension to inelastic collisions in multiple dimensions.
    • We will consider an example problem, illustrated in , in which one mass ($m_1$) slides over a frictionless surface into another initially stationary mass ($m_2$).
    • where $v_1$ is the initial velocity of the first mass, $v{}'_1$ is the final velocity of the first mass, $v_2$is the initial velocity of the second mass, and $\theta{}'_1$ is the angle between the velocity vector of the first mass and the x-axis.
    • Momentum is equal to the product of mass and velocity.
    • This illustrates the example problem in which one mass collides into another mass that is initially stationary.