Question

The moment of inertia of a solid about an arbitrary line \(L\) is defined as $$ I_{L}=\iiint_{R} d^{2} f(x, y, z) d x d y d z, $$ where \(f\) is density and \(d\) is the distance from a general point \((x, y, z)\) of the solid to the line \(L\). Prove the Parallel Axis theorem: $$ I_{L}=I_{\bar{L}}+M h^{2} . $$

Short answer

Question: Prove the Parallel Axis Theorem, which states that the moment of inertia of a solid about an arbitrary line L is equal to the moment of inertia about a parallel line passing through the center of mass, plus the mass times the square of the distance between the two lines. Solution: The proof of the Parallel Axis Theorem can be achieved by following these steps: 1. Calculate the moment of inertia about the parallel axis, \(\bar{L}\), passing through the center of mass. 2. Relate the distance from the point P to both axes, L and \(\bar{L}\). 3. Substitute the new expression into the expression for the moment of inertia about axis L. 4. Expand and simplify the resulting expression. 5. Show that one of the terms in the expression is equal to zero. 6. Combine the terms to arrive at the Parallel Axis Theorem, which is \(I_{L} = I_{\bar{L}} + Mh^2\).

Step-by-step solution

Calculate the moment of inertia about the parallel axis

To find the moment of inertia about the parallel axis, \(\bar{L}\), let's consider an arbitrary point P in the solid with coordinates (x, y, z). We denote the displacement vector from the center of mass to point P as \(\vec{r}\), and the distance vector from the point P to the parallel axis \(\bar{L}\) as \(\vec{d}_{\bar{L}}\). Then the distance from P to \(\bar{L}\) is \(d_{\bar{L}} = |\vec{d}_{\bar{L}}|\). The moment of inertia about the axis \(\bar{L}\) can be expressed as: $$ I_{\bar{L}} = \iiint_{R} d_{\bar{L}}^2 f(x, y, z) dx\,dy\,dz $$

Relate the distance from the point P to both axes

Consider the displacement vector from the center of mass of the solid to the point P, \(\vec{r}\). Now express the distance vector from point P to axis L, \(\vec{d}_L\), in relation to the distance vector from point P to \(\bar{L}\), \(\vec{d}_{\bar{L}}\), the distance vector from the L to the center of mass, \(\vec{h}\) and the displacement vector \(\vec{r}\): $$ \vec{d}_L = \vec{d}_{\bar{L}} - \vec{h} $$ Square both sides of this expression and remember that \(d^2_L = |\vec{d}_L|^2\) and \(d^2_{\bar{L}} = |\vec{d}_{\bar{L}}|^2\) $$ d^2_L = d^2_{\bar{L}} - 2\vec{h} \cdot \vec{d}_{\bar{L}} + h^2 $$

Substitute the new expression for the moment of inertia about the axis L

Now that we have found a relation between the distance from point P to both axes, we can substitute this into the expression for the moment of inertia about axis L: $$ I_{L} = \iiint_{R} (d^2_{\bar{L}} - 2\vec{h} \cdot \vec{d}_{\bar{L}} + h^2) f(x, y, z) dx\,dy\,dz $$

Expand and simplify the expression

Let's now expand and simplify the expression for the moment of inertia about axis L: $$ I_{L} = \iiint_{R} d^2_{\bar{L}} f(x, y, z) dx\,dy\,dz - 2 \iiint_{R} (\vec{h} \cdot \vec{d}_{\bar{L}}) f(x, y, z) dx\,dy\,dz + \iiint_{R} h^2 f(x, y, z) dx\,dy\,dz $$ By comparing this expression with the expression for \(I_{\bar{L}}\), we notice that the first term on the right-hand side is equal to \(I_{\bar{L}}\): $$ I_{L} = I_{\bar{L}} - 2 \iiint_{R} (\vec{h} \cdot \vec{d}_{\bar{L}}) f(x, y, z) dx\,dy\,dz + \iiint_{R} h^2 f(x, y, z) dx\,dy\,dz $$

Show the second term is equal to zero

Since \(\vec{h}\) is constant, it can be brought outside the integrals: $$ - 2 \vec{h} \cdot \iiint_{R} \vec{d}_{\bar{L}} f(x, y, z) dx\,dy\,dz $$ Now notice that the inner integral is equal to the center of mass of the solid: $$ - 2 \vec{h} \cdot \vec{\bar{R}}_M = 0 $$ Where \(\vec{\bar{R}}_M\) is the position vector of the center of mass.

Combine terms to arrive at the Parallel Axis Theorem

Finally, we can combine the expressions to obtain the Parallel Axis Theorem: $$ I_{L} = I_{\bar{L}} + \iiint_{R} h^2 f(x, y, z) dx\,dy\,dz $$ Rewriting the last term in the form \(Mh^2\) where M is the mass of the solid: $$ I_{L} = I_{\bar{L}} + Mh^2 $$ This proves the Parallel Axis Theorem.

Key concepts

Moment of Inertia

The concept of "moment of inertia" is an essential aspect of physics, especially in rotational dynamics. It is the rotational equivalent of mass for linear motion. Just as mass measures an object's resistance to changes in its state of motion along a straight path, the moment of inertia, usually denoted by the symbol \(I\), measures an object's resistance to changes in its rotational motion around an axis. The formula for the moment of inertia involves integrating over the volume of a solid:

• \(I = \iiint_{R} r^2 f(x, y, z) \, dx \, dy \, dz\)
• where \(r\) is the perpendicular distance from the axis of rotation to the element of mass, and \(f(x, y, z)\) is the density function.

This calculation is essential for solving problems in engineering and physics, such as determining the stress distribution in structural elements or the stability of rotating systems. Understanding the moment of inertia helps in designing objects to ensure that they function correctly in both rotation and linear motion.

Center of Mass

The center of mass is a crucial concept in physics and engineering that refers to the point where the entire mass of an object or system can be considered to be concentrated. For a solid object, the center of mass is determined mathematically by averaging the positions of all the mass elements of the object, weighted by their respective masses. It is essentially the "balance point" of an object.

• Mathematically, the center of mass \((x_{cm}, y_{cm}, z_{cm})\) is calculated by:
• \(x_{cm} = \frac{\iiint_{R} x \cdot f(x, y, z) \, dx \, dy \, dz}{M}\)
• \(y_{cm} = \frac{\iiint_{R} y \cdot f(x, y, z) \, dx \, dy \, dz}{M}\)
• \(z_{cm} = \frac{\iiint_{R} z \cdot f(x, y, z) \, dx \, dy \, dz}{M}\)
• where \(M\) is the total mass of the solid.

The position of the center of mass affects how the object responds to forces and torques. For instance, it determines the pivot point about which an object will rotate when subjected to a torque.

Density Function

A density function describes how mass is distributed throughout an object. It is a cornerstone in calculating moments of inertia and centers of mass, as it provides the relationship between volume and mass.

• Commonly denoted as \(f(x, y, z)\), the density function is used in integration over the volume of the object.
• The total mass \(M\) of the object is obtained by the integral: \(M = \iiint_{R} f(x, y, z) \, dx \, dy \, dz\).

In practical applications, knowing the density distribution helps in solving problems related to strength, stability, and deformation of materials. For example, in engineering, non-uniform density distributions might be designed to enhance an object’s mechanical properties.

Solid Mechanics

Solid mechanics is the branch of mechanics that studies the behavior of solid materials, especially their motion and deformation under forces. This discipline is fundamental in designing and analyzing objects such as buildings, bridges, aircraft, and machinery.

• It involves understanding stress, strain, and the material's elastic, plastic, and fracture responses to loads.
• Solid mechanics also covers different kinds of materials like metals, ceramics, polymers, and composites.

Practitioners of solid mechanics apply concepts such as moments of inertia and centers of mass to predict how objects will respond to various forces. This understanding is crucial for creating safe and reliable structures and machines. The Parallel Axis Theorem, which they frequently use, helps in calculating the moment of inertia for different rotational axes, facilitating better design and analysis.