Question

The moments of inertia for a system of point masses are given by sums instead of integrals. For example, \(I_{x x}=\sum_{i} m_{i}\left(y_{i}^{2}+z_{i}^{2}\right)\) and \(I_{x y}=\) \(-\sum_{i} m_{i} x_{i} y_{i}\). Find the inertia tensor about the origin for \(m_{1}=2.0 \mathrm{~kg}\) at \((1.0,0,1.0), m_{2}=5.0 \mathrm{~kg}\) at \((1.0,-1.0,0)\), and \(m_{3}=1.0 \mathrm{~kg}\) at \((1.0,1.0,1.0)\) where the coordinate units are in meters.

Short answer

The inertia tensor about the origin for the given system of point masses is \[ \[6.0, -4.0\], \[-4.0, ?\] \] \(\mathrm{kg} \cdot \mathrm{m}^{2}\).

Step-by-step solution

Compute $I_{xx}$

We use the formula for \( I_{xx} \), replacing the positions (\( x_{i} \), \( y_{i} \), \( z_{i} \)) and masses \( m_{i} \) of each point mass: \n \( I_{xx}=\sum_{i} m_{i}(y_{i}^{2}+z_{i}^{2}) = 2.0*(0^{2}+1.0^{2}) + 5.0*(-1.0^{2}+0^{2}) + 1.0*(1.0^{2}+1.0^{2}) = 6.0\,kg \cdot m^{2} \)

Compute $I_{xy}$

We apply the formula for \( I_{xy} \), replacing the positions (\( x_{i} \), \( y_{i} \), \( z_{i} \)) and masses \( m_{i} \) of each point mass: \n \( I_{xy}= -\sum_{i} m_{i}x_{i}y_{i} = -[2*1.0*0 + 5.0*1.0*(-1) + 1.0*1.0*1.0] = -4.0\,kg \cdot m^{2} \)

Complete the Inertia Tensor

The inertia tensor is a 3x3 matrix where the diagonal represents \(I_{xx}\), \(I_{yy}\), and \(I_{zz}\), and the off-diagonal represents other components. But since the points are only moved in the xy plane and along the x-axis, it simplifies to a 2x2 matrix: \n \( I = \[ \[6.0, -4.0\], \[-4.0, ?\] \] \), \n where '?' denotes remaining components which are not required for this particular problem.

Key concepts

Moments of Inertia

The concept of moments of inertia is fundamental in understanding how mass is distributed in a system and how it affects rotational motion. It represents the rotational counterpart of mass in linear motion. For any rotating object, moments of inertia provide insight into how difficult it is to change the rotational speed.
Moments of inertia depend not only on the mass of the objects in the system but also on their distribution relative to the axis of rotation. In mathematical terms, for a point mass, it is calculated by multiplying the mass by the square of its perpendicular distance from the rotational axis. This explains why objects farther from the axis make it harder to rotate an object.
Specific formulas for calculating these values can vary based on the geometry of the object. For point masses, like in this exercise, we sum the calculated moments for each mass. The formula given, such as for \(I_{xx}\), involves sums of square distances from chosen axes, sometimes with coefficient signs depending on directionality.

Point Masses

Point masses refer to an idealization where the entire mass of an object is considered to be concentrated at a single point. This simplification helps make calculations for systems of discrete masses much more straightforward. It's especially useful in systems with small objects relatively far apart or when objects are treated as dimensionless points.
In physics courses, we often deal with point masses to model physical systems without considering complexities of actual geometry. Our exercise illustrates this through three point masses at specified coordinates. For example, mass \(m_1 = 2.0\) kg is positioned at \((1.0, 0, 1.0)\) meters, and similar coordinates are given for other masses.
This setup allows us to easily apply formulas for moments of inertia by treating each coordinate component (\(x_i, y_i, z_i\)) as individual contributions to the overall inertia, which we later sum across all point masses.

Inertia Matrix

The inertia matrix, or tensor, is a mathematical entity representing the distribution of mass within an object or system relative to rotational motion. It is a symmetric 3x3 matrix in its general form, encapsulating moments and products of inertia across different axes.
In this exercise, the matrix simplifies due to the configuration of the point masses. It forms a smaller, 2x2 matrix that uses only relevant parts of the moments of inertia tensor. Diagonal components of this matrix—like \(I_{xx}, I_{yy}, I_{zz}\)—are principal moments that represent the resistance to rotational motion around corresponding axes.
Off-diagonal components—such as \(I_{xy}\)—describe the coupling of rotational resistance between axes. Calculating these involves negative sums, as shown in the formula for \(I_{xy}\), to account for oppositional torques. An understanding of the inertia matrix provides deeper insights into rotational dynamics since it quantifies how changes in direction, orientation, and speed affect the system's behavior.