Question

A rigid body consists of three equal masses $(m)$ fastened at the positions $(a,0,0),(0,a,2a)$ and $(0,2a,a).$ (a) Find the inertia tensor $\mathbf{I}$. (b) Find the principal moments and a set of orthogonal principal axes.

Short answer

Inertia tensor: $$m\left[\begin{array}{ccc}5a^2& 0& 0\\ 0& 6a^2& -4a^2\\ 0& -4a^2& 7a^2\end{array}\right]$$ Principal moments: $11a^2,7a^2,0$, with corresponding eigenvectors as principal axes.

Step-by-step solution

Understand the Problem

We need to calculate the inertia tensor for a system of three equal masses located at specified positions and then find the principal moments and axes of inertia for the system. The inertia tensor helps describe how the mass is distributed relative to an axis.

Recall the Inertia Tensor Formula

The inertia tensor for a set of point masses is given by: $$\mathbf{I}=\sum _i{m}_i\left[\begin{array}{ccccccc}{y}_i^2+z_i^2& -x_i{y}_i& -x_i{z}_i-y_i{x}_i& x_i^2+z_i^2& -y_i{z}_i-z_i{x}_i& -z_i{y}_i& x_i^2+y_i^2\end{array}\right],$$where the summation is taken over all point masses of the rigid body.

Calculate Moments of Inertia for Each Mass

Define the position vectors for the masses: - Mass at $(a,0,0)$- Mass at $(0,a,2a)$- Mass at $(0,2a,a)$Calculate the moment of inertia for each position and sum them up:- For the first mass at $(a,0,0)$, $$I_1=m\left[\begin{array}{ccccccc}{0}^2+0^2& 0\cdot a& 0\cdot 00\cdot a& a^2+0^2& 0\cdot 00\cdot 0& 0\cdot 0& a^2+0^2\end{array}\right]=m\left[\begin{array}{ccccccc}0& 0& 00& a^2& 00& 0& a^2\end{array}\right].$$- For the second mass at $(0,a,2a)$, $$I_2=m\left[\begin{array}{ccccccc}{a}^2+(2a{)}^2& 0\cdot a& 0\cdot 2a0\cdot a& 0^2+(2a{)}^2& -a\cdot 2a0\cdot 2a& -a\cdot 2a& 0^2+a^2\end{array}\right]=m\left[\begin{array}{ccccccc}5a^2& 0& 00& 4a^2& -2a^{2}0& -2a^2& 2a^2\end{array}\right].$$- For the third mass at $(0,2a,a)$, $$I_3=m\left[\begin{array}{ccccccc}(2a{)}^2+a^2& 0\cdot 2a& 0\cdot a0\cdot 2a& 0^2+a^2& -2a\cdot a0\cdot a& -2a\cdot a& 0^2+4a^2\end{array}\right]=m\left[\begin{array}{ccccccc}5a^2& 0& 00& a^2& -2a^{2}0& -2a^2& 4a^2\end{array}\right].$$

Sum the Components to Get the Inertia Tensor

Sum all three inertia matrices to obtain the total inertia tensor:$$\mathbf{I}=I_1+I_2+I_3=m\left[\begin{array}{ccccccc}5a^2& 0& 00& (a^2+4a^2+a^2)& (-2a^2-2a^2)0& (-2a^2-2a^2)& (a^2+2a^2+4a^2)\end{array}\right],$$Resulting in:$$\mathbf{I}=m\left[\begin{array}{ccccccc}5a^2& 0& 00& 6a^2& -4a^{2}0& -4a^2& 7a^2\end{array}\right].$$

Diagonalize the Inertia Tensor

To find the principal moments, we need to diagonalize the inertia tensor. Solve the characteristic equation $det(\mathbf{I}-\lambda {\mathbf{I}}_{\mathbf{3}})=0$, where ${\mathbf{I}}_{\mathbf{3}}$ is the identity matrix.$$m\left[\begin{array}{ccccccc}5a^2-\lambda & 0& 00& 6a^2-\lambda & -4a^{2}0& -4a^2& 7a^2-\lambda \end{array}\right]$$The characteristic equation is:$${\lambda }^3-18a^2{\lambda }^2+69a^4\lambda -104a^6=0$$Solve for $\lambda$ to find the principal moments.

Solve for Principal Moments and Axes

Solving the characteristic equation, we find the principal moments: $${\lambda }_1=11a^2,{\lambda }_2=7a^2,{\lambda }_3=0.$$ Corresponding eigenvectors give the principal axes. The orthogonal principal axes are derived using the standard procedure for finding eigenvectors.

Key concepts

Principal Moments of Inertia

The principal moments of inertia are special values that show how a rigid body resists angular acceleration around specific axes, known as the principal axes. These axes are perpendicular to each other and pass through the body's center of mass. Essentially, they are a particular set of axes for which the inertia tensor becomes diagonal, meaning all off-diagonal components become zero. This simplification makes it easier to analyze the rotational dynamics of the body.

For the given problem, we find the principal moments by solving the characteristic equation derived from the inertia tensor. The values of these principal moments (${\lambda }_1=11a^2,{\lambda }_2=7a^2,{\lambda }_3=0\lambda$), tell us about the rotational inertia of the body around these principal axes. If one of these moments is zero, it means the axis through the center of mass is an axis of rotation without resistance, indicating a principal axis lies along it.

Principal Axes

Principal axes are critical to understanding the rotational behavior of a rigid body. They are the axes around which the body can rotate without experiencing any torque due to inertia. These axes correspond to the eigenvectors of the inertia tensor. To find them, one must solve the eigenvector equation for the tensor.

In practical terms, these axes are the natural directions a body will rotate around if perturbed. For symmetric objects, like a cube or a sphere, the principal axes align with the object's geometric symmetry axes. In the calculation, by determining the principal moments first, we then find the orientation of these axes by solving the system that arises from substituting each principal moment into the inertia tensor equation. This gives us a set of orthogonal vectors corresponding to the principal axes.

Rigid Body Dynamics

Rigid body dynamics involves analyzing how solid objects move when subjected to forces. Unlike fluid or elastic bodies, a rigid body is assumed to have no deformation under force, maintaining its size and shape. The core elements of rigid body dynamics include translational and rotational motion, described by Newton's laws applied to mass and inertia.

The inertia tensor plays a critical role in describing how a rigid body resists angular motion. It quantitatively describes how mass distribution affects the body's rotation about an axis. In our specific exercise, we calculated this tensor for three point masses. By simplifying the dynamics into principal moments and axes, we make the analysis straightforward by reducing the inertia's complex interaction to simpler, more intuitive terms.

Inertia Tensor Calculation

The inertia tensor is a matrix that provides detailed information about how a body's mass is distributed relative to an axis of rotation. The formula for calculating the inertia tensor involves summing contributions from all point masses or particles composing the rigid body.

In the exercise, we calculated the inertia tensor by using the positions of the three masses. Each position contributed to the final tensor through its squared distances from each axis, as well as the products of its coordinates which fill the off-diagonal elements of the tensor. Once all contributions are summed, we can examine this matrix to extract physical properties like principal moments and principal axes, simplifying further analyses of the body's rotational dynamics. This process is crucial in many areas such as mechanical engineering and robotics, where understanding and controlling motion is essential.