Problem 49 - Thomas Calculus

Question

Proof of the Parallel Axis Theorem a. Show that the first moment of a thin flat plate about any line in the plane of the plate through the plate's center of mass is zero. (Hint: Place the center of mass at the origin with the line along the \(y\) -axis. What does the formula \(\overline{x}=M_{y} / M\) then tell you?) b. Use the result in part (a) to derive the Parallel Axis Theorem. Assume that the plane is coordinatized in a way that makes \(L_{\mathrm{cm}}\) . the \(y\) -axis and \(L\) the line \(x=h .\) Then expand the integrand of the integral for \(I_{L}\) to rewrite the integral as the sum of integrals whose values you recognize.

Step-by-Step Solution

Answer

The first moment is zero by symmetry, leading to the Parallel Axis Theorem: \( I_L = I_{\text{cm}} + Mh^2 \).

Step 1: Understanding the First Moment

The first moment of a body about a line is the product of its mass and its distance from the line. For a thin plate with its center of mass at the origin and the line along the y-axis, the first moment about this line is zero because the distribution of mass is symmetric about this axis. Thus, \( \overline{x}=\frac{M_y}{M} = 0 \) implies that the sum of moments about the y-axis (where \( M_y \) is the moment calculated as \( \int x \, dm \)) equals zero.

Step 2: Expressing Moment of Inertia

The moment of inertia of a shape about any axis L, parallel to an axis through the center of mass (\( I_{\text{cm}} \)), is calculated by considering the perpendicular distance \( h \) from the center of mass. The formula for the parallel axis theorem is \( I_L = I_{\text{cm}} + Mh^2 \).

Step 3: Coordinate Setup

Shift coordinates such that the center of mass (CM) is at the origin, and the y-axis is used as \( L_{\text{cm}} \). The line \( L \) is at \( x = h \). This places our line some distance \( h \) from the center of mass along the x-axis.

Step 4: Moment of Inertia Calculation

The moment of inertia about the line \( L \) (at \( x = h \)) is \( I_L = \int (x-h)^2 \, dm \). Expanding this, it becomes \( I_L = \int (x^2 - 2xh + h^2) \, dm \).

Step 5: Separating the Integral

Split the integral into three parts: \( I_L = \int x^2 \, dm - 2h \int x \, dm + h^2 \int dm \). Recognize that \( \int x^2 \, dm \) is the moment of inertia about the y-axis through the CM and \( \int x \, dm = 0 \) due to symmetry (from Step 1).

Step 6: Final Simplification

With \( \int x \, dm = 0\), the formula simplifies to \( I_L = I_{\text{cm}} + Mh^2 \), where \( h^2 \int dm = Mh^2 \) is added for the parallel distance squared. This completes the proof of the parallel axis theorem.

Key Concepts

Moment of Inertia

The moment of inertia is a critical concept in physics and engineering, as it helps to describe how difficult it is to change the rotational motion of an object. Think of it as the rotational equivalent of mass in linear motion. It is denoted by the symbol \( I \) and is calculated based on the distribution of mass and the axis around which the object rotates. In this context, the center of mass plays a significant role because it helps define the axis through which we measure this property.
A basic formula for the moment of inertia about an axis is given by the integral:

• \( I = \int r^2 \, dm \)

Here, \( r \) is the perpendicular distance from the axis to a small element of mass \( dm \). For a line through the center of mass, the moment of inertia is at its minimum, which simplifies many aspects of analysis.Understanding the moment of inertia is crucial when applying the Parallel Axis Theorem, which relates the moment of inertia about an axis through the center of mass \( (I_{\text{cm}}) \) to that about another parallel axis \( (I_L) \) with a formula:

• \( I_L = I_{\text{cm}} + Mh^2 \)

where \( M \) is the total mass and \( h \) is the perpendicular distance between the two axes.

Center of Mass

The center of mass (CM) is a point that acts as if the entire mass of the body system is concentrated at that spot for the purpose of motion analysis. In simple terms, it is the balancing point of an object, where all parts perfectly counterbalance each other.
When analyzing the motion or applying theorems like the Parallel Axis Theorem, setting the center of mass at the origin of your coordinate system often simplifies the calculation. This locates the origin at the geometric

• Average of all the mass distribution in the system

For a given body, if you set the CM at the origin, your calculations involving rotational properties become more straightforward. When working with the first moment which involves the center of mass, we note:

• The moment about any line through the center of mass is zero
  

This comes from the symmetry of mass distribution around the CM. As a result, this property simplifies calculations, proving essential in deriving the Parallel Axis Theorem.

Coordinate System

In physics, setting up a coordinate system is crucial for solving problems effectively. By choosing the right system, we can simplify our equations and make use of symmetries within the problem. Often, aligning one of the axes with an important part of the problem, like the center of mass or the axis of rotation, is advantageous.
When dealing with the Parallel Axis Theorem, we typically:

• Place the center of mass at the origin of the coordinate system
• Align the axis of interest along a coordinate axis, such as the y-axis

This allows the calculations to become more streamlined. For instance, once the CM is at the origin, the moments about this point can be studied directly using the specified coordinate axis. Also, we can keep track of distances, like \( h \), from the CM to any parallel axis in a straightforward manner.

Integration

Integration is a mathematical technique needed to calculate quantities like the moment of inertia in continuous mass distributions. It is essentially about summing up an infinite number of infinitesimally small elements to find a total. In mechanical physics, this often means adding up all small mass elements \( dm \) in systems with complex geometries.
For the Parallel Axis Theorem, integration is used to determine how mass is distributed relative to chosen axes. By integrating over the body's volume or area, you can find quantities such as the moment of inertia, where:

• \( \int x^2 \, dm \)
• \( \int x \, dm \)

This process helps in expressing the moment of inertia \( I_L \), about an axis not through the center of mass, in terms of known quantities like \( I_{\text{cm}} \).
Integrals can sometimes be simplified by recognizing them as standard forms (like symmetry properties), where known results can be directly used, significantly simplifying our work when deriving complex theorems such as the Parallel Axis Theorem.