PX155 - B2 - continuous bodies

more common to treat bodies as continuous, not a collection of distinct points
- eg: $N \to \infty$ and $m_{i} \to 0$ infinite number of points(elements) of mass $d m$ $$\vec r_{cm}=\frac{\int \vec r dm}{\int dm}$$
  - NB: $\int \vec{r} d m = (\int x d m, \int y d m, \int z d m)$
  - x-component of ${\vec{r}}_{c m} = \frac{\int x d m}{\int d m}$
eg: a bar of length $l$ has mass per unit length $ρ = α x$ , where $x$ is the distance from one end
- to find: the position of its centre of mass

M = \int d m = \int_{0}^{L} ρ . d x = \int_{o}^{L} α x . d x = α [\frac{x^{2}}{2}]_{0}^{L} = \frac{α L^{2}}{2}

- similarly,$$\int x.dm=\int_0^L x\rho.dx =\int_o^L\alpha x^2 .dx = \alpha[\frac{x^3}{3}]_0^L= \frac{\alpha L^3}{3}$$
- hence,

x_{c m} = \frac{\int x . d m}{\int d m} = \frac{2}{3} L

σ = \frac{M}{A} ⟹ M = \frac{π R^{2}}{2} σ (= \int d m)

- slicing the semicircle vertically:
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