Question

The uniform solid block in Fig 10-38has mass 0.172kg and edge lengths a = 3.5cm, b = 8.4cm, and c = 1.4cm. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces.

Short answer

The rotational inertia about an axis through one corner and perpendicular to the large faces is, $4.7\times {10}^{-4}kg.m^2$.

Step-by-step solution

Listing the given quantities

Masses and coordinates of four particles are

1. The mass of block is, M = 0.172kg.
2. The edge length are, a = 3.5cm, b = 3.5cm, and c = 1.4cm.

Concept and Formula used for the given question

By using the concept of moment of inertia we can calculate the moment of inertia about its axis. For the slab, if the axis is not passing through the center, then we can use the parallel axis theorem to calculate the moment of inertia.

1. Fortheslab when axis is passing through its center then, ${\mathit{l}}_{\mathbf{c}\mathbf{o}\mathbf{m}}\mathbf{=}\frac{\mathbf{M}}{\mathbf{12}}\left(a^2+b^2\right)$
2. Whentheaxis is not passing through its center, then according totheparallel axis theorem,
   $\mathit{l}\mathbf{=}{\mathit{l}}_{\mathbf{c}\mathbf{o}\mathbf{m}}\mathbf{+}\mathit{m}{\mathit{h}}^{\mathbf{2}}$
   

Calculation of rotational inertia about an axis through one corner and perpendicular to the large faces

For solving parallel axis theorem, wecalculate the perpendicular distance from the axis passing through its center and corner.

$h=\sqrt{{\left(\frac{a}{2}\right)}^2+{\left(\frac{b}{2}\right)}^2}$

According to the parallel axis theorem,

localid="1661141288241" $\begin{array}{rcl}l& =& l_{com}+M{h}^2\\ & =& \frac{M}{12}\left(a^2+b^2\right)+M\left[{\left(\frac{a}{2}\right)}^2+{\left(\frac{b}{2}\right)}^2\right]\\ & =& \frac{M}{12}\left(a^2+b^2\right)+M\left[{\left(a\right)}^2+{\left(b\right)}^2\right]\\ l& =& \frac{M}{3}\left(a^2+b^2\right)\end{array}$

Substitute all the value in the above equation.

$\begin{array}{rcl}l& =& \frac{0.172kg}{3}\left[{\left(3.5\times {10}^{-2}m\right)}^2+{\left(8.4\times {10}^{-2}m\right)}^2\right]\\ & =& 4.7\times {10}^{-4}kg.m^2\end{array}$

Therefore, rotational inertia about an axis through one corner and perpendicular to the large faces is, $4.7\times {10}^{-4}kg.m^2$.