Question

The following notation is used in the problems: \(M=\) mass, \(\bar{x}, \bar{y}, \bar{z}=\) coordinates of center of mass (or centroid if the density is constant), \(I=\) moment of inertia (about axis stated), \(I_{x}, I_{y}, I_{z}=\) moments of inertia about \(x, y, z\) axes, \(I_{m}=\) moment of inertia (about axis stated) through the center of mass. Note: It is customary to give answers for \(I, I_{m}, I_{x},\) etc., as multiples of \(M\) (for example, \(I=\frac{1}{3} M l^{2}\) ). Prove the "parallel axis theorem": The moment of inertia I of a body about a given axis is \(I=I_{m}+M d^{2},\) where \(M\) is the mass of the body, \(I_{m}\) is the moment of inertia of the body about an axis through the center of mass and parallel to the given axis, and \(d\) is the distance between the two axes.

Short answer

The theorem is proven: I = I_m + M d^2.

Step-by-step solution

Define the variables and the problem

We need to prove the parallel axis theorem. Given variables are:- Let the mass of the body be denoted by M.- The moment of inertia about the center of mass axis is denoted by I_m.- The distance between two parallel axes is d.The theorem states that the moment of inertia I about the given axis is related to I_m by the formula: I = I_m + M d^2.

Set up the coordinate system

Consider a mass element dm located at a point with coordinates (x, y) in the plane. The center of mass (COM) is at (\bar{x}, \bar{y}). The distance from this element to the center of mass axis is \[ r_{COM}^2 = (x - \bar{x})^2 + (y - \bar{y})^2. \]

Express the moment of inertia about the center of mass axis

The moment of inertia about the axis through the center of mass (I_m) is given by the integral: \[ I_m = \int r_{COM}^2 dm = \int ((x - \bar{x})^2 + (y - \bar{y})^2) dm. \]

Express the moment of inertia about the given axis

The moment of inertia I about the given axis parallel to the COM axis, but at a distance d from it, is given by: \[ I = \int r_{d}^2 dm = \int ((x - \bar{x} + d_x)^2 + (y - \bar{y} + d_y)^2) dm, \] where \( d_x \) and \( d_y \) are the distances between the two axes.

Expand the integrants

Expand the integrands: \[ (x - \bar{x} + d_x)^2 = (x - \bar{x})^2 + 2(x - \bar{x}) d_x + d_x^2, \] \[ (y - \bar{y} + d_y)^2 = (y - \bar{y})^2 + 2(y - \bar{y}) d_y + d_y^2. \]

Combine and simplify the integrals

Combining the above, we get: \[ I = \int ((x - \bar{x})^2 + 2 (x - \bar{x}) d_x + d_x^2 + (y - \bar{y})^2 + 2 (y - \bar{y}) d_y + d_y^2) dm. \] This can be split into three separate integrals: \[ I = \int ((x - \bar{x})^2 + (y - \bar{y})^2) dm + \int 2 ((x - \bar{x}) d_x + (y - \bar{y}) d_y) dm + \int (d_x^2 + d_y^2) dm. \]

Simplify each integral

By definition, the center of mass coordinates (\bar{x}, \bar{y}) average out individual terms (x - \bar{x}) and (y - \bar{y}) over the mass distribution, so those cross terms will sum to zero: \[ \int 2 ((x - \bar{x}) d_x + (y - \bar{y}) d_y) dm = 0. \] The third integral is straightforward: \[ \int (d_x^2 + d_y^2) dm = (d_x^2 + d_y^2) \int dm = M (d_x^2 + d_y^2). \]

Combine the results

Putting all together: \[ I = I_m + M (d_x^2 + d_y^2). \] If the distance d between the two axes in a given direction is such that \( d^2 = d_x^2 + d_y^2 \), we obtain the desired formula: \[ I = I_m + M d^2. \]

Key concepts

moment of inertia

The moment of inertia is a fundamental concept in mechanics. It measures how much resistance a body has to rotational motion around a specific axis.
The formula for the moment of inertia, usually denoted as I, depends on the body’s shape and the distribution of its mass.
For example, the moment of inertia for a point mass (m) at a distance (r) from the rotation axis is given by: \( I = m r^2 \).
In general, the moment of inertia is calculated using integrals to sum up the contributions of all mass elements (dm) of a body:
\( I = \int r^2 \, dm \), where r is the distance from the axis of rotation.

center of mass

The center of mass (COM) is a point representing the average position of the mass of a body or system.
In simpler terms, it’s the balance point of the body where it can be completely supported.
If the mass is uniformly distributed, the center of mass coincides with the centroid.
The coordinates of the center of mass are determined using the following formulas for a system of particles:

• \( \bar{x} = \dfrac{1}{M} \sum m_i x_i \)
• \( \bar{y} = \dfrac{1}{M} \sum m_i y_i \)
• \( \bar{z} = \dfrac{1}{M} \sum m_i z_i \)

Here, \( M \) is the total mass, and \( m_i \) and \( x_i, y_i, z_i \) are the masses and coordinates of each particle.
For continuous mass distributions, integrals replace summations.

coordinate system

A coordinate system is essential for analyzing rotational motion. It helps in defining the position of mass elements and the axes involved.
Typically, a Cartesian coordinate system (with x, y, z axes) is used.
The choice of the coordinate system and the positioning of the origin can simplify calculations.
In the parallel axis theorem, we often choose a coordinate system where:

• the center of mass of the body is at the origin, i.e., (\( \bar{x} = 0, \bar{y} = 0, \bar{z} = 0 \)).
• the axes of rotation are parallel and one of them passes through the center of mass.

This setup simplifies calculations and helps in deriving relations such as the parallel axis theorem.

mass distribution

The distribution of mass within a body significantly affects its moment of inertia.
Mass distribution describes how mass is spread out in space within an object.
Whether mass is concentrated at the center or distributed towards the edges impacts the resistance to rotation.
Irregular objects require integrals to account for all variations in mass distribution.
For any mass element (dm) at coordinates (x, y, z), its contribution to the moment of inertia is influenced by its position.
Therefore, the overall moment of inertia is an integral of the form \( I = \int r^2 \, dm \), where \( r \) is the distance of the mass element from the rotation axis.
Understanding mass distribution is also crucial when applying the parallel axis theorem, which adjusts the moment of inertia when the rotation axis moves parallel to itself but away from the center of mass.