Mass of a wire, moments and center of mass.

Consider the curve $\mathcal{C}$as a thin wire; if we know at each point what is the density of the material used to build this wire, we can compute the mass, the moments of theis object about the coordinate planes, the center of mass of the object, and so on. We denote by $\delta (x,y,z)$the density at the point (x,y,z).

1.

Mass:

$$M= \int_{\mathcal{C}} \delta (x,y,z) \; ds.$$

2.

First moments about the coordinate planes:
$$\begin{align*}M_{xy} & = \int_{\mathcal{C}} z\delta (x,y,z) \; ds\\ M_{yz} & = ... ...x,y,z) \; ds\\ M_{zx} & = \int_{\mathcal{C}} y\delta (x,y,z) \; ds \end{align*}$$

3.

Coordinates of the center of mass:

$$\overline{x}=\frac {M_{yz}}{M} \qquad ; \qquad \overline{y}=\frac {M_{xz}}{M} \qquad ; \qquad \overline{z}=\frac {M_{xy}}{M}$$

4.

Moments of inertia:
$$\begin{align*}I_x &= \int_{\mathcal{C}} (y^2+z^2)\delta (x,y,z) \; ds\\ I_y &= ... ...(x,y,z) \; ds\\ I_L &= \int_{\mathcal{C}} r^2 \delta (x,y,z) \; ds \end{align*}$$
where L is any line in the space and r is the distance from the point (x,y,z) to the line L.

5.

Radius of gyration about the line L:

$$R_L= \sqrt{\frac {I_L}{M}}.$$

Example 8.7   Let $(x,y,z)= (\cos 2t, \sin 2t , 3t, \; 0 \leq t \leq 2 \pi$be a parameterization of $\mathcal{C}$(this curve is displayed on Fig.  3. Suppose that the density is uniformly equal to 1.

 

  

Figure 3: Two loops of a wire.

 

• Mass:
  $$\begin{align*}M& = \int_{\mathcal{C}} 1 \; ds \\ \quad & = \int_{0}^{2 \pi } \s... ...\\ \quad & = \ int_{0}^{2 \pi } \sqrt{10} \; dt = 2 \pi \sqrt{10}. \end{align*}$$
• First moments about the coordinate planes:
  $$\begin{align*}M_{xy} & = \int_{\mathcal{C}} z \; ds\\ \quad &= \int_{0}^{2 \pi ... ...} \; dt = \sqrt{10} \left[ -\frac 12 \cos 2t \right]_0^{2 \pi} =0. \end{align*}$$
• Coordinates of the center of mass:

$$\overline{x}=\frac {M_x}{M}=0 \qquad ; \qquad \overline{y}=\f... ...c {M_z}{M}=\frac {6 \pi ^2 \sqrt{10}}{2 \pi \sqrt{10}} =3 \pi.$$