Question

Let $R=(O,\mathcal{T})$ be a rest frame of a rigid body. Denote the entries in the corresponding inertia matrix by ${\mathcal{J}}_{ij}$. Show that the moment of inertia about an axis through $O$ in the direction of a unit vector with components $x_i$ is ${\mathcal{J}}_{ij}{x}_i{x}_j.$

Short answer

Answer: The expression is $I={\mathcal{J}}_{11}{x}_1^2+{\mathcal{J}}_{22}{x}_2^2+{\mathcal{J}}_{33}{x}_3^2+2{\mathcal{J}}_{12}{x}_1{x}_2+2{\mathcal{J}}_{13}{x}_1{x}_3+2{\mathcal{J}}_{23}{x}_2{x}_3$.

Step-by-step solution

Define the given variables

We are given a rigid body in a rest frame $R=(O,\mathcal{T})$ with the inertia matrix ${\mathcal{J}}_{ij}$. An axis passes through the point O, in the direction of a unit vector with components $x_i$. The goal is to find the moment of inertia about this axis.

Recall the formula for the moment of inertia

The moment of inertia about an axis can be found using the following formula: $I={\mathcal{J}}_{ij}{x}_i{x}_j$ Here, $I$ is the moment of inertia, and the indices $i$ and $j$ run from 1 to 3, representing the three dimensions.

Apply the formula to the given problem

We can now apply the formula for the moment of inertia to our given problem. Since $x_i$ are the components of a unit vector in the direction of the axis, we can write: $I={\mathcal{J}}_{ij}{x}_i{x}_j$

Expand the expression for $I$

To find the moment of inertia, we need to compute the multiplication of the inertia matrix elements with the components of the unit vector squared. The indices $i$ and $j$ run from 1 to 3: $I={\mathcal{J}}_{11}{x}_1^2+{\mathcal{J}}_{22}{x}_2^2+{\mathcal{J}}_{33}{x}_3^2+2{\mathcal{J}}_{12}{x}_1{x}_2+2{\mathcal{J}}_{13}{x}_1{x}_3+2{\mathcal{J}}_{23}{x}_2{x}_3$

Interpret the result

The moment of inertia about an axis through point O in the direction of a unit vector with components $x_i$ is given by the expression we derived: $I={\mathcal{J}}_{11}{x}_1^2+{\mathcal{J}}_{22}{x}_2^2+{\mathcal{J}}_{33}{x}_3^2+2{\mathcal{J}}_{12}{x}_1{x}_2+2{\mathcal{J}}_{13}{x}_1{x}_3+2{\mathcal{J}}_{23}{x}_2{x}_3$ This expression shows how the moment of inertia depends on the components of the inertia matrix and the unit vector in the direction of the axis.

Key concepts

Rigid Body Dynamics

Rigid body dynamics involves the study of solid objects, assuming they do not deform when forces are applied. In this context, a rigid body is a system of particles whose relative distances remain constant over time. This means that the position of particles within the body does not change, even though the entire object may move or rotate.

To properly understand rigid body dynamics, it's crucial to consider its two main types of motion:

• Translational motion: Here, every point in the rigid body moves in parallel lines, such as a car driving straight down a street. The body moves from one location to another without rotating around its center of mass.
• Rotational motion: This is when the body spins around a specific axis. Each part of the object moves along a circular path centered at the axis of rotation.

Describing rotational motion requires understanding the orientation of the body in space, typically with respect to a reference frame. Frames like the one mentioned in the exercise allow us to analyze how the body behaves under various forces and torques.

Inertia Matrix

When trying to understand rotational movement, the inertia matrix becomes essential. It is a 3x3 matrix that represents how mass is distributed throughout a rigid body. This matrix's elements, typically denoted as ${\mathcal{J}}_{ij}$, illustrate the resistance the body offers to rotational acceleration about different axes.

The inertia matrix can be thought of as a generalized moment of inertia. While the moment of inertia is scalar when dealing with simple objects rotating about a single axis, the inertia matrix can handle more complex situations.
This matrix is especially useful in 3D dynamics, as it accounts for:

• Mass distribution along each axis $(i,j)$
• Rotational symmetry of the body
• Principal axes of rotation that simplify calculations

Understanding the inertia matrix allows engineers and physicists to predict how forces and torques affect objects' rotation, especially when these objects have irregular shapes. By working with an inertia matrix, we can make deductions about the rotations about multiple intersecting axes simultaneously.

Unit Vector

A unit vector is a vector of length one that indicates direction without specifying magnitude. It is a standardized format often used in physics and engineering to simplify calculations involving directions.

In this context, the unit vector is pivotal while determining the moment of inertia about a specific axis. When the problem states "a unit vector with components $x_i$", it implies that:

• The vector's magnitude is normalized to one.
• The vector components ($x_1,x_2,x_3$) indicate the direction of the axis in a 3D space.

Unit vectors are beneficial when analyzing motion since they allow decoupling of direction and magnitude. Naturally, multiplying any vector by a unit vector projects it along that direction, which is essential when calculating moments of inertia around a specific, defined axis. Utilizing unit vectors ensures that the computed moment of inertia aligns with the direction we are interested in, enabling more accurate and simplified computations.