91Ó°ÊÓ

The density of a substance is a measurement of how densely it is packed together. The mass per unit volume is how it's defined.

Consider the figure, as given below –

We integrate an even function over an even interval, so –

$$\begin{gathered} M=∫dm \\ =∫\mathrm{λ}ds \\ ={∫}_{-1}^1\overline{)x}\overline{)}\sqrt{1+y^r}dx \\ ={∫}_{-1}^1\overline{)x}\overline{)}\sqrt{1+4x^2}dx \\ =2{∫}_{-1}^{1}x\sqrt{1+4x^2}dx \\ x=\frac{1}{2}\sin h^{2}udx \\ =\frac{1}{2}\cos h^{2}udu \\ =\frac{1}{2}{∫}_0^{\sin h^{-1}2}\cos h^{2}u\sin hudu \\ =\frac{1}{2}{∫}_0^{\sin h^{-1}2}\cos h^{2}ud\left(\cos hu\right) \\ ={\overline{)\frac{1}{6}\cos h^{3}u}}_0^{\sin h^{-1}2} \\ =\frac{1}{6}\left[\cos h^3\left(\sin h^{-1}2\right)-1\right]≈1.697 \end{gathered}$$

Therefore, the mass of the chain is obtained as 1.697.

The value ofandis calculated as –

$\begin{array}{l}y=\frac{1}{M}∫y\left(x\right)\mathrm{λ}ds\\ =\frac{1}{M}{∫}_{-1}^1{x}^2\left|x\right|\sqrt{1+4x^2}dx\\ =\frac{2}{M}{∫}_0^1{x}^3\sqrt{1+4x^2}dx\end{array}$

Here again the integration of an even function is done over an even interval. The same substitution is used as above –

$$\begin{gathered} \overline{y}=\frac{1}{8M}{∫}_0^{\sin h^{-1}2}\sin h^{3}u\cos h^{2}uudu \\ =\frac{1}{8M}{∫}_0^{\sin h^{-1}2}\left(\cos h^{2}u-1\right)\cos h^{2}ud\left(\cos hu\right) \\ =\frac{1}{8m}{\overline{)\left[\frac{1}{5}\cos h^{5}u-\frac{1}{3}\cos h^{3}u\right]}}_0^{\sin h^{-1}2} \\ ≈\frac{0.948}{M}≈0.559 \end{gathered}$$

Therefore, the value of$\left(\overline{x},\overline{y}\right)$ coordinates of centre of mass is obtained as (0,0.559).