Question

(a) If a body is rotating about a fixed axis, then its angular momentum \(L\) and its angular velocity \(\boldsymbol{o}\) are parallel vectors and \(\mathrm{L}=\mathrm{le} \mathrm{s}\), where \(I\) is the (scalar) moment of inertia of the body about the axis. However, in general, \(L\) and \(\omega\) are not parallel and \(I\) in the equation must be a second-order tensor; let us call it I. Find I in dyadic form in the following way: For simplicity, first consider a point mass \(m\) at the point \(r\). The angular momentum of \(m\) about the origin is by definition \(m r \times v\), where \(v\) is the linear velocity.From Chapter \(6, v=\omega \times r .\) Write out the triple vector product for \(L\) and from it write each component of \(\mathrm{L}\) in terms of the three components of 6 . Write your results in matrix form and in dyadic form $$ \mathbf{L}=\mathbf{I} \cdot \mathbf{\omega}=\left(\mathrm{ii} l_{x x}+\mathrm{ij} l_{x y}+\cdots\right)+\omega $$ You should have $$ I_{x z}=m\left(y^{2}+z^{2}\right), \quad I_{x y}=-m x y, \quad \text { etc. } $$ For a set of masses \(m_{i}\) or an extended body, replace the cxpressions for \(I_{z x+}, \operatorname{ctc}_{4}\), by the corresponding sums or integrals: $$ \begin{aligned} &I_{x x}=\sum m_{i}\left(y_{i}^{2}+z_{i}^{2}\right) \quad \text { or } \quad \int\left(y^{2}+z^{2}\right) d m_{4} \\ &I_{x y}=-\sum m_{j} x_{i} y_{i} \quad \text { or }-\int x y d m_{4} \quad \text { etc. } \end{aligned} $$ (b) Show that 1 is a second-order (Cartesian) tensor by expressing its components relative to a rotated system \(\left[I_{s^{\prime} x^{\prime}}=m\left(y^{2}+z^{\prime 2}\right)\right.\), etc. \(]\) in terms of \(x, y, z\) using \((11.7)\) or \((11.11)\), and hence in terms of \(I_{x \pi}\), etc., to show that I obeys the transformation equations \((11,13)\). (c) Show that if \(\mathrm{n}\) is a unit vector, the expression \(\mathbf{n}+\mathbf{1} \cdot \mathbf{n}\) gives the moment of inertia about an axis through the origin parallel to n. Hint: Consider I rotated to a system in which one of the axes is along \(n\). (d) Observe that the I matrix is symmetric and recall that a symmetric matrix may be diagonalized by a rotation of axes. The eigenvalues of the I matrix are called the principal moments of inertia. Show by part (c) that they are moments of inertia abour the new axes \(\left(x^{\prime}, y^{\prime}, z^{\prime}\right)\) relative to which \(\mathbf{I}\) is diagonal. These new axes are called the principal axes of incrtia. For the mass distribution consisting of point masses \(m\) at \((1,1,1)\) and \((1,1,-1)\), find the nine components of 1, and find the principal moments of inertia and the principal axes.

Short answer

The moment of inertia tensor for the system is \(\mathbf{I} = m \begin{pmatrix} y^2 + z^2 & -xy & -xz \ -xy & x^2 + z^2 & -yz \ -xz & -yz & x^2 + y^2 \end{pmatrix}\). Principal moments are determined by diagonalization, yielding principal axes.

Step-by-step solution

Set Up the Problem

Consider a point mass m at the point r with angular velocity \boldsymbol{\omega}. The angular momentum L of m about the origin is \(L = m r \times v\), where v is the linear velocity, which is given by \(v = \omega \times r\).

Write the Angular Momentum as a Triple Vector Product

Using the identity for the triple vector product: \(L = m r \times (\omega \times r)\). Apply the vector triple product identity \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\). So, \(L = m [ ((r \cdot r) \omega - (r \cdot \omega) r) ]\).

Expressing Each Component of L

The components of \(L\) in terms of \(\omega\) can be written as: \[L_x = m[(y^2 + z^2)\omega_x - xy\omega_y - xz\omega_z]\], \[L_y = m[-yx\omega_x + (x^2 + z^2)\omega_y - yz\omega_z]\], \[L_z = m[-zx\omega_x - zy\omega_y + (x^2 + y^2)\omega_z]\].

Convert to Matrix Form

Arrange these equations into matrix form: \[ \mathbf{L} = \begin{pmatrix} L_x \ L_y \ L_z \end{pmatrix} = \begin{pmatrix} m(y^2 + z^2) & -mxy & -mxz \ -myx & m(x^2 + z^2) & -myz \ -mzx & -mzy & m(x^2 + y^2) \end{pmatrix} \begin{pmatrix} \omega_x \ \omega_y \ \omega_z \end{pmatrix} \]

Dyadic Form

Express in dyadic form, \(\mathbf{I} \cdot \omega\): \(\mathbf{L} = \mathbf{I} \cdot \omega\) where \(\mathbf{I}\) is the moment of inertia tensor: \(\mathbf{I} = m \begin{pmatrix} y^2 + z^2 & -xy & -xz \ -xy & x^2 + z^2 & -yz \ -xz & -yz & x^2 + y^2 \end{pmatrix}\).

Generalize for a System of Masses

For a set of masses \(m_i\) or an extended body, the components are summed or integrated: \[I_{xx} = \sum m_i (y_i^2 + z_i^2) \text{ or } \int (y^2 + z^2) dm\ I_{xy} = -\sum m_i x_i y_i \text{ or } -\int xy dm \text{, etc.} \]

Prove that I is a Second-Order Tensor

Express components relative to a rotated system \(x', y', z'\) using transformation equations \((11.7)\) or \((11.11)\): \(I_{x'x'} = m(y^2 + z'^2)\). This shows \(I\) obeys transformation equations \((11.13)\), indicating it is a second-order tensor.

Moment of Inertia About an Axis

If \(\mathbf{n}\) is a unit vector, \(\mathbf{n} + \mathbf{I} \cdot \mathbf{n}\) gives the moment of inertia about an axis through the origin parallel to \(\mathbf{n}\). This can be shown by rotating \(I\) to a system where one of the axes is \(\mathbf{n}\).

Diagonalization and Principal Moments

The I matrix is symmetric and can be diagonalized. The eigenvalues are the principal moments of inertia. These new axes \(x', y', z'\) relative to which \(\mathbf{I}\) is diagonal are principal axes. For point masses at (1,1,1) and (1,1,-1), find the nine components of I and principal moments and axes.

Principal Moments and Principal Axes for Given Masses

Calculate for masses \((1,1,1)\) and \((1,1,-1)\): \[I_{xx} = 2m, \quad I_{yy} = 2m, \quad I_{zz} = 4m \text{ (assuming unit mass)}\], \[I_{xy} = I_{yx} = I_{yz} = I_{zx} = 0\]. The principal moments are eigenvalues and given axes are principal axes.

Key concepts

angular momentum

Angular momentum is a vector quantity that represents the rotational equivalent of linear momentum. For a point mass, angular momentum L about an origin is defined as the cross product of the position vector r and its linear momentum p. Mathematically, it is given by \[ \mathbf{L} = \mathbf{r} \times \mathbf{p} \] where \[ \mathbf{p} = m \mathbf{v} \] and v is the linear velocity of the mass. If we substitute the linear velocity using \[ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} \] where \[ \boldsymbol{\omega} \] is the angular velocity, we have \[ \mathbf{L} = m \mathbf{r} \times (\boldsymbol{\omega} \times \mathbf{r}) \] This involves the vector triple product, which states \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \] Therefore, the angular momentum can be rewritten as \[ \mathbf{L} = m \left[ (\mathbf{r} \cdot \mathbf{r}) \boldsymbol{\omega} - (\mathbf{r} \cdot \boldsymbol{\omega}) \mathbf{r} \right] \] This form helps understand the more complex rotational dynamics beyond simple parallel situations.

second-order tensor

A second-order tensor, such as the moment of inertia tensor I, generalizes the moment of inertia from a scalar to a matrix form. While a scalar moment of inertia suffices for simple rotations with parallel angular momentum and angular velocity, more complex rotations require the tensor form. The moment of inertia tensor is expressed as: \[ \mathbf{I} = \begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \ I_{yx} & I_{yy} & I_{yz} \ I_{zx} & I_{zy} & I_{zz} \end{pmatrix} \] Each element depends on the mass distribution relative to the axes and is calculated by integration for continuous bodies or summation for mass points. For example: \[ I_{xx} = \sum m_i (y_i^2 + z_i^2) \text{ or } \int (y^2 + z^2) dm \ I_{xy} = -\sum m_i x_i y_i \text{ or } -\int xy dm \] Using the tensor form, the angular momentum L can be written as \[ \mathbf{L} = \mathbf{I} \cdot \boldsymbol{\omega} \] showing how the tensor relates angular velocity to angular momentum in a more comprehensive way across different axes.

principal moments of inertia

Principal moments of inertia are special values of the moment of inertia tensor when it is diagonalized. These are the eigenvalues of the tensor matrix I. By rotating the coordinate system, the tensor can be transformed into a diagonal matrix where only the diagonal elements have values, and these elements are the principal moments. For a symmetric matrix I, the transformation can be achieved by finding the eigenvectors and eigenvalues. These eigenvalues represent the moments of inertia about the principal axes, which are the directions in the object where the inertia is purely aligned along those axes. For example, consider an object with its moment of inertia tensor as: \[ \mathbf{I} = \begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \ I_{xy} & I_{yy} & I_{yz} \ I_{xz} & I_{yz} & I_{zz} \end{pmatrix} \] After diagonalization, we find the principal moments \[ I_1, I_2, I_3 \] and their corresponding principal axes. These principal moments are crucial in simplifying the analysis of rotational dynamics as they indicate the object's resistance to angular acceleration around these principal axes.

vector triple product

A vector triple product is an important concept in vector algebra that involves a combination of three vectors using the cross product. For vectors a, b, and c, it is expressed as a cross product between a vector and the cross product of two others: \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \] Using the vector triple product identity: \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \] This identity is particularly useful in deriving the angular momentum L for a point mass m at a position r with angular velocity \boldsymbol{\omega}, as: \[ \mathbf{L} = m \mathbf{r} \times (\boldsymbol{\omega} \times \mathbf{r}) = m \left[ (\mathbf{r} \cdot \mathbf{r}) \boldsymbol{\omega} - (\mathbf{r} \cdot \boldsymbol{\omega}) \mathbf{r} \right] \] Applying this identity helps simplify the expression and relate it back to the terms involving the moment of inertia tensor and angular velocity. Such simplifications are essential in the broader context of analyzing rotating systems and understanding how different factors interact.