moment of inertia

Examples of moment of inertia in the following topics:

• Applications of Multiple Integrals

  • As is the case with one variable, one can use the multiple integral to find the average of a function over a given set.
  • Given a set $D \subseteq R^n$ and an integrable function $f$ over $D$, the average value of $f$ over its domain is given by:
  • where $m(D)$ is the measure of $D$.
  • In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
  • 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:

• 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.
  • In the case of a system of particles $P_i, i = 1, \cdots , n$, each with mass $m_i$ that are located in space with coordinates $\mathbf{r}_i, i = 1, \cdots , n$, the coordinates $\mathbf{R}$ of the center of mass satisfy the condition:
  • where $M$ is the sum of the masses of all of the particles.
  • Use multiple integrals to find the center of mass of a distribution of mass

• The Mean Value Theorem, Rolle's Theorem, and Monotonicity

  • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
  • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
  • To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour.
  • The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).
  • For any function that is continuous on $[a, b]$ and differentiable on $(a, b)$ there exists some $c$ in the interval $(a, b)$ such that the secant joining the endpoints of the interval $[a, b]$ is parallel to the tangent at $c$.