Question

Consider a rigid plane body or "lamina," such as a flat piece of sheet metal, rotating about a point \(O\) in the body. If we choose axes so that the lamina lies in the \(x y\) plane, which elements of the inertia tensor \(\mathbf{I}\) are automatically zero? Prove that \(I_{z z}=I_{x x}+I_{y y}\).

Short answer

The zero elements are \(I_{xz}, I_{zx}, I_{yz}, \) and \(I_{zy}\). We proved \(I_{zz} = I_{xx} + I_{yy}\).

Step-by-step solution

Understanding the Problem

We need to determine which elements of the inertia tensor \(\mathbf{I}\) are zero for a rigid plane body (lamina) lying in the \(xy\) plane. Additionally, we must prove that the diagonal element \(I_{zz}\) is equal to \(I_{xx} + I_{yy}\).

Identifying Zero Elements of the Inertia Tensor

For a lamina in the \(xy\) plane, the mass is distributed along these two axes, meaning there is no mass in the \(z\)-axis direction. Thus, any moments of inertia involving the \(z\)-axis with \(x\) or \(y\) will be zero. Therefore, \(I_{xz} = I_{zx} = I_{yz} = I_{zy} = 0\).

Expression of the Inertia Tensor Elements

In addition to zero elements identified earlier, the inertia tensor has the form: \[\mathbf{I} = \begin{bmatrix}I_{xx} & I_{xy} & 0 \I_{yx} & I_{yy} & 0 \0 & 0 & I_{zz} \\end{bmatrix}.\] Due to symmetry and the properties of the inertia tensor, \(I_{xy} = I_{yx}\).

Relating Inertia Tensor Elements to Mass Distribution

For the lamina, the sum of the diagonal entries, also known as the trace of the tensor, gives the same information as the sum of masses around the principal axes. Since no mass lies along the \(z\)-axis (the normal to the lamina), the inertia associated with \(I_{zz}\):\[ I_{zz} = I_{xx} + I_{yy}. \] This relationship is due to how inertia distributes in a plane perpendicular to the axis through the point \(O\).

Cross-Verify the Relationship

Since the trace of the inertia tensor should be invariant, consider a coordinate rotation to verify that the total inertia for any axis perpendicular to the plane sums to the trace. Thus, the inertia associated by the plane along \(z\)-axis (normal to the plane) is the sum of \(I_{xx}\) and \(I_{yy}\).

Solution Conclusion

Thus, we conclude that the zero elements of the inertia tensor \(\mathbf{I}\) for a lamina in the \(xy\) plane are \(I_{xz}, I_{zx}, I_{yz}, I_{zy}\), and we have shown that \(I_{zz} = I_{xx} + I_{yy}\).

Key concepts

Rigid Body Mechanics

Rigid body mechanics is a branch of classical mechanics dealing with the behavior of solid objects that don’t deform when forces are applied. This assumption makes analysis simpler since it disregards factors like stress and strain that affect more pliable bodies.
In the context of our exercise, it’s essential as it allows us to consider the geometric and inertial properties of objects such as laminas, which are infinitely thin layers. These objects can be represented in fixed, unchanging shapes, making calculations straightforward.
Since the assumptions hold that distances between any two points in the rigid body always remain constant, we can easily define properties like the center of mass and moment of inertia. For analyzing rotation dynamics, a rigid body is particularly suitable due to these simplifications, facilitating the determination of the inertia tensor.

Moment of Inertia

The moment of inertia is a fundamental concept in the description of rotating bodies. It measures an object's resistance to changes in its angular velocity. For a simple analogy, think of it as the rotational equivalent of mass in linear motion.
In mathematical terms, it’s often represented by the inertia tensor, which is particularly useful for complex bodies. Each element in the tensor reflects mass distribution relative to specific axes of rotation.

• The higher the moment of inertia, the harder it is to change the rotation of the body about that particular axis.
• In our scenario, where the body lies in the xy-plane, the inertia tensor simplifies as there are no components related to the z-axis.

Understanding the reduction of some tensor elements to zero, as derivable from the symmetry and mass disposition in our exercise, is key to analyzing plane bodies.

Lamina

A lamina refers to a thin, flat plane body with a simple geometric shape, like a sheet of metal or paper.

• Because it is two-dimensional and lies in the xy-plane, it does not extend into the third dimension, the z-axis.
• This reduces the complexity of its inertia tensor, assuming uniform thickness and material distribution.

Such objects are traditionally easy to work with due to their simplified mass distribution, focusing only on the plane they occupy. Knowing that the z-axis influence is negligible helps capitalize on symmetry properties while working with calculations such as inertia tensors. In the context of our exercise, the zero elements of the inertia tensor reflect this lack of mass distribution in the z-direction.

Plane Body

A plane body in physics refers to an object that occupies a flat two-dimensional surface. These bodies, like a lamina, are strictly confined within a plane, most commonly the xy-plane.
For any plane body, defining mass distribution becomes straightforward since it primarily concerns only two dimensions.
This simplicity translates to practical advantages in calculation, especially when dealing with inertia tensors or rotational dynamics.

• Such simplification allows for evaluating rotational inertia without concern for out-of-plane components.
• The horizontal orientation ensures properties related to the normal axis, which doesn’t feature any mass from the body extending into it, showcasing the advantages when evaluating the inertia tensor.

The ease with which properties like mass moment location and rotation resistance can be assessed underpins why plane bodies are a fundamental study element in rigid body mechanics.