Problem 11 - An Introduction to Mechanics

Question

Tensor of inertia A particle of mass \(m\) is located at \(x=2, y=0, z=3\). (a) Find its moments and products of inertia relative to the origin. (b) The particle undergoes pure rotation about the \(z\) axis through a small angle \(\alpha\). Show that its moments and products of inertia are unchanged to first order in \(\alpha\) if \(\alpha \ll 1\).

Step-by-Step Solution

Answer

Moments of inertia are \(I_x = 9m\), \(I_y = 13m\), \(I_z = 4m\); products are \(I_{xy} = 0\), \(I_{xz} = -6m\), \(I_{yz} = 0\). Rotation doesn't change them to first order if \(\alpha \ll 1\).

Step 1: Define moments of inertia at the origin

For a point mass located at coordinates \((x, y, z)\), the moments of inertia relative to the origin about the axes are given by \( I_x = m(y^2 + z^2) \), \( I_y = m(x^2 + z^2) \), and \( I_z = m(x^2 + y^2) \). The products of inertia are \( I_{xy} = -mxy \), \( I_{xz} = -mxz \), and \( I_{yz} = -myz \).

Step 2: Calculate initial moments of inertia

For the given particle with mass \(m\) located at \((x=2, y=0, z=3)\), substitute these values into the formulas: \[ I_x = m(0^2 + 3^2) = 9m \] \[ I_y = m(2^2 + 3^2) = 13m \] \[ I_z = m(2^2 + 0^2) = 4m \].

Step 3: Calculate initial products of inertia

Substitute the coordinates into the products of inertia formulas: \[ I_{xy} = -m(2)(0) = 0 \] \[ I_{xz} = -m(2)(3) = -6m \] \[ I_{yz} = -m(0)(3) = 0 \].

Step 4: Analyze changes due to rotation

The rotation about the \(z\)-axis by angle \(\alpha\) can be described by the coordinate transformation: \[ x' = x \cos(\alpha) - y \sin(\alpha) \approx x - y \alpha \] \[ y' = x \sin(\alpha) + y \cos(\alpha) \approx y + x \alpha \] For small \(\alpha\), \(\cos(\alpha) \approx 1\) and \(\sin(\alpha) \approx \alpha\). The dependences of \(z'\) on \(z\) stay unchanged since rotation about the \(z\) axis doesn’t affect the \(z\)-coordinate. Thus, in first order, the new coordinates \((x', y', z')\) are \((2, 0, 3)\), indicating inertia components remain unchanged.

Key Concepts

Understanding the Tensor of Inertia

When dealing with rotational dynamics, the tensor of inertia plays a crucial role in understanding how mass is distributed in relation to an axis of rotation. For a single mass point, the tensor of inertia comprises moments and products of inertia. These components tell us how difficult it is to start or stop the rotation of the mass about a particular axis.
In the exercise, we calculate the moments of inertia for a particle with mass \( m \) located at \( (x=2, y=0, z=3) \). The moments of inertia for this system are given by the equations:

• \( I_x = m(y^2 + z^2) \)
• \( I_y = m(x^2 + z^2) \)
• \( I_z = m(x^2 + y^2) \)

The terms \( I_x, I_y, \) and \( I_z \) represent resistance to rotation about the x, y, and z axes respectively.
Additionally, the products of inertia give information on how much the axes are skewed relative to each other. They are calculated as

• \( I_{xy} = -mxy \)
• \( I_{xz} = -mxz \)
• \( I_{yz} = -myz \)

These quantities help us understand the complex nature of rotational movements when multiple axes are involved.

The Role of Coordinate Transformation

Coordinate transformations are mathematical operations that bring clarity when observing objects in motion from different perspectives. When a rotation occurs, such as in this exercise around the \( z \) axis through a small angle \( \alpha \), we often need to use coordinate transformations to calculate changes accurately.
For a small angle \( \alpha \), we use approximations \( \cos(\alpha) \approx 1 \) and \( \sin(\alpha) \approx \alpha \).
This simplification leads to the new coordinates:

• \( x' = x - y\alpha \)
• \( y' = y + x\alpha \)
• \( z' = z \)

This approximation shows that small rotations about the \( z \) axis lead to minimal changes in the \( x \) and \( y \) directions, with the \( z \) coordinate remaining unchanged.
Thus, we comprehend that for infinitesimal angles, the tensor of inertia remains largely unaffected by simple rotations, providing a consistent description of inertia.

Exploring Pure Rotation Effects

Pure rotation implies that all parts of a system rotate about a specific axis without other types of motion occurring. It's often studied to simplify complex physical situations, much like in this instructional problem involving a small angle rotation around the \( z \) axis.
If a particle undergoes pure rotation about an axis, the tensor of inertia, to first order in small angle \( \alpha \), remains unchanged. This means that the calculated moments and products of inertia from the first steps are unaffected by the rotation if \( \alpha \ll 1 \) since the position of the particle relative to the origin does not substantially change.
This insight greatly aids in understanding the inherent properties of systems in rotational motion, allowing physics students to grasp the impact of rotational dynamics without the need for complicated calculations. Pure rotation simplifies the analysis by ensuring changes are minimal and within a predictable scope for very small rotations.