Question 1. Consider an homogeneous square plate of side length \(l\), thickness \(dz\) and mass center \(O\). The axes \(Ox\), \(Oy\) are parallel to the edges. Axis \(Oz\) is perpendicular to the plate. See Figure 1.
Find its moment of inertia \(dI_{Oz}\) about axis \(Oz\) as a function of \(l\), \(dz\), and its volumetric mass density \(\rho\) (consult a table).
Question 2. Now consider the body 1 shaped in the form of a square-based pyramid. See Figure 2. It is made of the same material as the plate. The side length of its base \(Oxy\) is \(a\) and its height is \(h\). Its mass is \(m = \rho a^2 h/3\) and its mass center \(G\) is at the height \(z_G = h/4\).
Question 2.1. Justify that the inertia matrix about \(G\) of this body on basis \((\boldsymbol{\hat{x}},\boldsymbol{\hat{y}},\boldsymbol{\hat{z}})\) is diagonal with \(I_{Gx}=I_{Gy}\).
Question 2.2. Using the result of question 1, find the moment of inertia \(I_{Oz}\) of body 1 about axis \(Oz\) as a function of \(m\), \(h\) and \(a\).
Question 2.3. After showing that \(J=\int z^2 dm = 3 m h^2/80\), find \(I_{Gx}\) (use symmetry to relate \(I_{Gx}\) to \(I_{Gz}\) and \(J\)).
Question 2.4. Find the inertia matrix about vertex \(C\) on basis \((\boldsymbol{\hat{x}},\boldsymbol{\hat{y}},\boldsymbol{\hat{z}})\).
Question 3. Body 1 is now used as a spinning top by mounting a stem (of negligible mass) along its axis \(Cz\). It is spun on a horizontal support so that its vertex \(C\) remains fixed. The inclination of its axis \((C,\boldsymbol{\hat{z}})\) relative to the vertical remains fixed. After parametrizing the orientation of body 1 using Euler angles, find the linear momentum and angular momentum \(\boldsymbol{H}_C\) of the body in terms of these angles.
© 2025 R. Valéry Roy.