expansion

Examples of expansion in the following topics:

• Thermal Stresses

  • Solids also undergo thermal expansion.
  • What are the basic properties of thermal expansion?
  • In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.
  • What is the underlying cause of thermal expansion?
  • Thermal stress is created by thermal expansion or contraction.

• Volume Expansion

  • The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. illustrates that, in general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions.
  • The subscript p indicates that the pressure is held constant during the expansion.
  • This is an expansion of 0.2%.
  • For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient.
  • Compare the effects of the pressure on the expansion of gaseous and solid materials

• Linear Expansion

  • Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.
  • Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.
  • The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion; it generally varies with temperature.
  • To a first approximation, the change in length measurements of an object (linear dimension as opposed to, for example, volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient.
  • Thermal expansion of long continuous sections of rail tracks is the driving force for rail buckling.

• Area Expansion

  • We learned about the linear expansion (in one dimension) in the previous Atom.
  • Objects expand in all dimensions, and we can extend the thermal expansion for 1D to two (or three) dimensions.
  • If the metal is heated, we can guess that the the piece, in general, will get larger due to thermal expansion.
  • For isotropic materials, and for small expansions, the linear thermal expansion coefficient is one half of the area coefficient.
  • Express the area thermal expansion coefficient in the form of an equation

• Free Energy and Work

  • The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system.
  • The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system.
  • "Useful" in this case, refers to the work not associated with the expansion of the system.
  • For instance, examples of useful, non-expansion work in biological organisms include muscle contraction and the transmission of nerve impulses.

• Finding a Specific Term

  • There may be instances when we want to identify a certain term in the expansion of $\displaystyle{ (x+y) }^{ n }$.
  • Let's go through a few expansions of binomials, in order to consider any patterns that are present in the terms.
  • However, what about longer expansions?
  • There is a shortcut for identifying particular terms of longer expansions.
  • The following formula yields the $r$th term in the expansion:

• Settlement of the New Land

  • The United States began continental expansion immediately after the Constitution of 1789 through war, treaty, land deals, and settlement.
  • In particular, these powers fought over western expansion, running from the Mississippi River to the Pacific.
  • However, with the success of the American Revolution, westward expansion and territorial acquisition of the North American continent became a U.S.
  • The continental expansion of the United States was mostly accomplished through treaty, purchase, or war with southern neighbors over the span of the nineteenth century.
  • Identify key dates in the history of the United States' territorial expansion

• The Business Cycle: Definition and Phases

  • Business cycles are identified as having four distinct phases: expansion, peak, contraction, and trough.
  • An expansion is characterized by increasing employment, economic growth, and upward pressure on prices.
  • The slowing ceases at the trough and at this point the economy has hit a bottom from which the next phase of expansion and contraction will emerge.
  • An expansion is the period from a trough to a peak, and a recession as the period from a peak to a trough.
  • The phases of a business cycle follow a wave-like pattern over time with regard to GDP, with expansion leading to a peak and then followed by contraction leading to a trough.

• Binomial Expansions and Pascal's Triangle

  • The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
  • Therefore, what follows is a shortcut for finding binomial expansions using a visual tool.
  • To understand how this pattern applies to the binomial formula, consider the expansion:
  • where the coefficients $a_i$ in this expansion are precisely the numbers on row $n$ of Pascal's triangle.
  • Applying these numbers to the binomial expansion, we have:

• The Rise of the West

  • Westward expansion was motivated by the Jeffersonian ideal of the yeoman farmer and enabled by technological improvements.
  • Westward expansion was sped by improvements in transportation infrastructure, which carried settlers westward.
  • Westward expansion was given further incentives by improved agricultural technology and better market access.
  • As American expansion continued, Native Americans resisted settlers' encroachment in several regions of the new nation.
  • Tecumseh led a Native American coalition that attempted to stop westward expansion of the United States.