PlanetPhysics: center of mass of half-disc

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center of mass of half-disc

(Example)

Let be the upper half-disc of the disc $x^2+y^2 \leqq R$ in $\mathbb{R}^2$ with a constant surface-density 1. By the symmetry, its centre of mass locates on its medium radius, and therefore we only have to calculate the ordinate of the centre of mass. For doing that, one can use in this two-dimensional case instead a triple integral the double integral

$\displaystyle Y = \frac{1}{\nu(E)}\int\!\!\int_E y\,dx\,dy,$

where $\nu(E) = \frac{\pi R^2}{2}$ is the area (and the mass) of the half-disc. The region of integration is defined by

$\displaystyle E = \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; -R\leqq x \leqq R,\; 0 \leqq y \leqq \sqrt{R^2-x^2}\}.$

Accordingly, we may write

$\displaystyle Y = \frac{2}{\pi R^2}\!\int_{-R}^R\!dx\int_0^{\sqrt{R^2-x^2}}\!y\... ...\!\!-R}^{\,\quad R}\left(\frac{R^2x}{2}-\frac{x^3}{6}\right) = \frac{4R}{3\pi}.$

Thus the centre of mass is the point $(0,\,\frac{4R}{3\pi})$ .

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See Also: centre of mass of polygon

This object's parent.

Cross-references: mass, centre of mass

This is version 3 of center of mass of half-disc, born on 2007-07-02, modified 2009-04-18.
Object id is 253, canonical name is CentreOfMassOfHalfDisc.
Accessed 813 times total.

Classification:

Physics Classification:	45.40.-f (Dynamics and kinematics of rigid bodies)
	45.50.Dd (General motion)
	02.40.Yy (Geometric mechanics )

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