Question

A chain in the shape $\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$ between $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{1}$ and$\mathit{x}\mathbf{=}\mathbf{1}$ has density. Find

(a) M,

(b)$\overline{\mathbf{x}}\mathbf{}\mathbf{,}\mathbf{}\overline{\mathbf{y}}$.

Short answer

(a) The mass of the chain in the shape $y=x^2$between x = -1 and x = 1, having density of is 1.697.

(b) The value of $\left(\overline{x},\overline{y}\right)$coordinates of centre of mass for a chain in the shape $y=x^2$between x = -1 and x = 1, having density of $\left|x\right|$is (0,0.559).

Step-by-step solution

Definition of density

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.

(a) Determining mass of the chain

Consider the figure, as given below –

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

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

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

(b) Determining x ,y coordinates of centre of mass

The value ofandis calculated as –

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

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

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

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