Problem 96 - University Physics with Modern Physics

Question

A thin, uniform rod is bent into a square of side length \(a\) . If the total mass is \(M\) , find the moment of inertia about an axis through the center and perpendicular to the plane of the square. (Hint: Use the parallel-axis theorem.)

Step-by-Step Solution

Answer

The moment of inertia is \( \frac{7M}{12}a^2 \).

Step 1: Understand the Problem

We are given a rod bent into a square of side length \(a\) with a total mass \(M\). We need to find the moment of inertia of this square about an axis through its center and perpendicular to its plane.

Step 2: Break Down the Problem

The square is made of four rods, each of length \(a\), with a mass of \(\frac{M}{4}\) since the mass is uniformly distributed. The moment of inertia will be calculated for each rod and then summed together to get the total moment of inertia for the square.

Step 3: Moment of Inertia for One Side about Its Own Center

The moment of inertia of a rod of length \(a\) and mass \(\frac{M}{4}\) about an axis through its center and perpendicular to its length is \(\frac{1}{12} \left(\frac{M}{4}\right)a^2\).

Step 4: Apply the Parallel Axis Theorem

We need the moment of inertia for each side around the axis through the center of the square. Using the parallel-axis theorem, \(I = I_{cm} + md^2\), where \(d\) is the distance from the center of the rod to the axis through the center of the square. Here, \(d = \frac{a}{\sqrt{2}}\). So, \(I = \frac{1}{12} \left(\frac{M}{4}\right)a^2 + \frac{M}{4}\left(\frac{a}{\sqrt{2}}\right)^2\).

Step 5: Calculate Each Rod's Moment of Inertia Using the Parallel Axis Theorem

Plug the values \(d = \frac{a}{\sqrt{2}}\) into the parallel axis formula for each rod:\[I = \frac{1}{12} \left(\frac{M}{4}\right)a^2 + \frac{M}{4}\frac{a^2}{2} = \frac{M}{48}a^2 + \frac{M}{8}a^2\]Simplify this to get \(I = \frac{7M}{48}a^2\) for each side.

Step 6: Total Moment of Inertia for All Four Sides

Since the moment of inertia is the sum of each side's moment of inertia, multiply the result of step 5 by 4:\[I_{total} = 4 \times \frac{7M}{48}a^2 = \frac{7M}{12}a^2\]

Step 7: Conclusion

The moment of inertia of the square about an axis through its center and perpendicular to its plane is \( \frac{7M}{12}a^2 \).

Key Concepts

Parallel Axis Theorem

The Parallel Axis Theorem is a powerful tool in physics and engineering. It allows us to calculate the moment of inertia of a body about any axis, given the moment of inertia about a parallel axis through the center of mass and the perpendicular distance between the axes.
In mathematical terms, the theorem states:

• \[ I = I_{cm} + md^2 \]

Here:

• \( I \) is the moment of inertia about the new axis.
• \( I_{cm} \) is the moment of inertia about the center of mass.
• \( m \) is the mass of the object.
• \( d \) is the distance between the two axes.

Using this theorem, we efficiently analyze complex systems by breaking them into simpler parts. For the square made of rods in the exercise, it simplifies the calculation by giving a method to adjust the moment of inertia for the square's center axis.

Uniform Rod

A uniform rod is a rod with its mass distributed evenly along its length. If broken into smaller sections, each part has the same mass per unit length. This is a significant simplification when calculating physical properties like the moment of inertia.
For a uniform rod of length \( a \) and total mass \( M \), the mass per unit length is \( \frac{M}{a} \). In our exercise, the uniform rod is bent into a square, which affects how we calculate properties like the center of mass and moment of inertia for each side.
Remember, bending the rod into a square does not change the total mass. It divides the mass equally among the four sides, each containing \( \frac{M}{4} \). Because the rod shares its properties evenly, calculations become straightforward when using the parallel axis theorem.

Axis Perpendicular to Plane

An axis perpendicular to a plane extends out of the plane, creating an angle of 90 degrees with it. Picture a pencil standing on a flat surface — the pencil represents the axis, and the surface represents the plane. In rotatory mechanics, calculating the moment of inertia around such an axis is common.
In this exercise, the axis runs through the center of the square and is perpendicular to its plane. This orientation of the axis is crucial because it determines how far each mass element of the square is from the axis, affecting the moment of inertia calculation.

• Here, symmetry helps simplify moment of inertia calculations by equally spacing elements around the center.

A perpendicular axis facilitates the use of symmetry, ensuring equal contribution from each mass segment distributed at an equal distance from the center.

Center of Mass

The center of mass is a point representing the average position of the distributed mass of a body. For uniform and symmetric bodies, it is located at their geometric center. In practical terms, it's where you could balance the object perfectly.
For our problem, the center of mass of the entire square is at its geometric center, directly in the middle of the square's plane. This point simplifies calculations, as each rod contributing to the square is equidistant from this point.

• The position of the center of mass affects both how bodies rotate and how their moment of inertia is determined.

In formulas, knowing the center aids in applying the parallel axis theorem as it pinpoints where length \( d \) of the rod from the center of mass to the new axis originates.