Infinitesimal Mass Elements
Infinitesimal Mass Elements - Related Videos
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Elements: Chemical Symbols and Isotopes
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2020
A chemical symbol is an abbreviation used to indicate an element or an atom of an element. For example, the symbol for mercury is Hg. The same symbol is used to indicate one atom of mercury (microscopic domain) or to label a container of many atoms of the element mercury (macroscopic domain). Some symbols are derived from the common English name of the element; others are abbreviations of the name in another language — Latin, Greek or German. For example, the symbol for aluminum (common name)...
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Periodic Classification of the Elements
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2020
The periodic table arranges atoms based on increasing atomic number so that elements with the same chemical properties recur periodically. When their electron configurations are added to the table, a periodic recurrence of similar electron configurations in the outer shells of these elements is observed. Because they are in the outer shells of an atom, valence electrons play the most important role in chemical reactions. The outer electrons have the highest energy of the electrons in an atom...
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Atomic Mass
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2020
Atoms — and the protons, neutrons, and electrons that compose them — are extremely small. For example, a carbon atom weighs less than 2 × 10−23 g. When describing the properties of tiny objects such as atoms, we use appropriately small units of measure, such as the atomic mass unit (amu). The amu was originally defined based on hydrogen, the lightest element, then later in terms of oxygen. Since 1961, it has been defined with regard to the most abundant isotope of carbon, atoms of which are...
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Classification of Elements and Compounds
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Pure substances consist of only one type of matter. A pure substance can be an element or a compound. An element consists of only one type of atom, while a compound consists of two or more types of atoms held together by a chemical bond. Elements are classified as atomic or molecular based on the nature of their basic units. Compounds are pure substances composed of two or more elements in fixed, definite proportions. Compounds are classified as ionic or molecular (covalent) based on the bonds...
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Molar Mass
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2020
The identity of a substance is defined not only by the types of atoms or ions it contains but by the quantity of each type of atom or ion. For example, water, H2O, and hydrogen peroxide, H2O2, are alike in that their respective molecules are composed of hydrogen and oxygen atoms. However, because a hydrogen peroxide molecule contains two oxygen atoms, as opposed to the water molecule, which has only one, the two substances exhibit very different properties. Atoms and molecules are extremely...
FAQs
To assign dm, a model pairs the material’s density with an appropriately sized differential measure: length for a line-like body, area for a surface, or volume for a three-dimensional body. The resulting element represents a local mass contribution, which can then be combined with contributions from neighboring portions without assuming the entire object has uniform density.
Nonuniform density changes the value of each local contribution, so equal-sized geometric portions need not carry equal mass. Infinitesimal mass elements preserve that variation throughout the body rather than replacing it with one average value. This matters when integration is used to determine total mass or other quantities that depend on how mass is distributed.
Compared with a discrete-particle model, this approach describes a continuous body through an idealized collection of differential contributions. That distinction is useful when the object has complex geometry or continuously varying density, because particle-by-particle bookkeeping would be impractical. The same framework can support mass, center-of-mass, moment-of-inertia, gravitational, and inertial calculations.
Begin by identifying the body’s geometry and density description, then select a differential length, area, or volume that matches the representation. Express dm using density and that measure, describe how the element spans the body, and integrate its contributions over the relevant region. This workflow accommodates both simple and complex shapes.
Once local contributions are expressed, integration can produce the body’s total mass and locate its center of mass. The same mass distribution also enters calculations of moments of inertia and gravitational or inertial effects. Thus, one setup connects geometric description and density to several mechanical properties of the body.
They are especially useful for rigid-body motion, fluid systems, and engineering structures when geometry or density makes a discrete-particle description impractical. In physics, they provide a calculus-based route from local mass distribution to whole-system behavior. In engineering analysis, the same representation helps organize calculations for bodies whose shape or material distribution varies across space.