Math 2.0

Real functions Definition

Centre of mass

Builds on Triple integralDouble integral

Definition

Let E⊆ℝnE \subseteq \mathbb{R}^n be a bounded Jordan measurable set, and let δ ⁣:E→[0,∞)\delta \colon E \to [0, \infty) be a continuous density with total mass M=∫Eδ(x) dx>0M = \int_E \delta(x) \, dx > 0. The centre of mass of EE with density δ\delta is the point xˉ=(xˉ1,…,xˉn)\bar{x} = (\bar{x}_1, \dots, \bar{x}_n) with xˉi=1M∫Exi δ(x) dx,i=1,…,n.\bar{x}_i = \frac{1}{M} \int_E x_i \, \delta(x) \, dx, \qquad i = 1, \dots, n. If δ\delta is constant, xˉ\bar{x} is the centroid of EE, and it does not depend on the value of the constant.

Remarks

For n=3n = 3 the integrals are triple integrals, and for a plane lamina, n=2n = 2, they are double integrals. For point masses m1,…,mkm_1, \dots, m_k at p1,…,pkp_1, \dots, p_k, the centre of mass is ∑jmjpj/∑jmj\sum_j m_j p_j / \sum_j m_j. The definition above replaces the sum by an integral. Example: the half disc {x2+y2≤a2, y≥0}\{x^2 + y^2 \le a^2, \ y \ge 0\} with constant density has centroid (0,4a/(3π))(0, 4a / (3\pi)). If EE and δ\delta are symmetric under a reflection, then xˉ\bar{x} lies in the mirror of that reflection.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Center_of_mass, A continuous distribution of mass
  2. Web https://en.wikipedia.org/wiki/Center_of_mass, A system of particles
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DefinesCentre of mass
UsesTriple integral, Double integral

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