Problem 38 - College Physics

Question

Two objects of different masses are joined by a light rod. (a) Is the moment of inertia about the center of mass the minimum or the maximum? Why? (b) If the two masses are \(3.0 \mathrm{~kg}\) and \(5.0 \mathrm{~kg}\) and the length of the rod is \(2.0 \mathrm{~m},\) find the moments of inertia of the system about an axis perpendicular to the rod, through the center of the rod and the center of mass.

Step-by-Step Solution

Answer

(a) Minimum, as CM is point of equilibrium. (b) Center of mass: \(7.5\) kg·m², Center of rod: \(8.0\) kg·m².

Step 1: Understanding Moment of Inertia

The moment of inertia ( In") is a measure of how much a shape resists rotational motion about a given axis. It depends on the mass distribution relative to the axis. The center of mass is usually the axis of minimum inertia for a given object because masses are distributed symmetrically about this point, minimizing the inertia.

Step 2: Calculate Center of Mass

The center of mass (CM) can be found using the formula \( x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \). Here, choose \( x_1 = 0 \) at the position of the first mass, and \( x_2 = 2 \) for the second. So, \( x_{CM} = \frac{3 \times 0 + 5 \times 2}{3 + 5} = 1.25 \) m from the first mass.

Step 3: Determine Moment of Inertia Through Center of Mass

Using the parallel axis theorem, the moment of inertia about the CM is calculated using \( I = m_1d_1^2 + m_2d_2^2 \), where \( d_1 \) and \( d_2 \) are the distances from each mass to the CM. Thus, \( I = 3 \times (1.25)^2 + 5 \times (0.75)^2 = 3 \times 1.5625 + 5 \times 0.5625 = 4.6875 + 2.8125 = 7.5 \) kg·m².

Step 4: Determine Moment of Inertia Through Center of the Rod

For the axis through the center of the rod, use the parallel axis theorem: \( I = I_{CM} + Md^2 \) where \( M = 3 + 5 = 8 \) kg and \( d = 0.25 \) m (distance from CM to center of rod). Thus, \( I = 7.5 + 8 \times (0.25)^2 = 7.5 + 8 \times 0.0625 = 8.0 \) kg·m².

Key Concepts

Understanding Center of Mass

The center of mass is a critical concept when dealing with rotational motion and moments of inertia. It's essentially the point where you can consider the entire mass of a system to be concentrated. This point gives you an important reference for analyzing motion and calculating moments of inertia.
For a simple system, like our two-object example, the center of mass lies somewhere between the two masses, influenced by their weights. If one mass is heavier, the center of mass will be closer to it. In our example, with masses of 3.0 kg and 5.0 kg separated by a rod of 2.0 m, the center of mass was calculated to be 1.25 m from the 3 kg mass.
Understanding the center of mass allows physicists to simplify complex systems and make calculations more manageable. In our specific problem, it helped determine the moment of inertia about different axes.

Exploring the Parallel Axis Theorem

The parallel axis theorem is a useful tool in rotational dynamics. It's used to calculate the moment of inertia of a body around any axis, given the moment of inertia around a parallel axis that goes through the center of mass. This is especially helpful when the axis of rotation isn't aligned with the center of mass.
To apply this theorem, you use the formula: \[ I = I_{CM} + Md^2 \]where:

• \( I \) is the moment of inertia about the new axis,
• \( I_{CM} \) is the moment of inertia about the center of mass,
• \( M \) is the total mass,
• \( d \) is the distance between the two axes.

In our exercise, we used this theorem to find the moment of inertia about an axis through the rod's center, which turned out to be 8.0 kg·m². It provided a straightforward way to relate different moments of inertia without re-evaluating the entire mass distribution.

Analyzing Rotational Motion

Rotational motion involves the movement of bodies around an internal or external axis. It is a key area in physics because many real-world systems rotate rather than move linearly. Understanding rotational motion requires a grasp of the concept of torque and inertia. Torque is the force that causes the object to rotate, while inertia—which in rotational terms is the moment of inertia—resists that change.
The moment of inertia itself provides insight into how easy or difficult it is to change the object's rotational state. Large moments of inertia imply it's hard to start or stop rotation, similar to how mass impacts linear motion. In our exercise, we analyzed how the distribution of an object's mass affects its moment of inertia around various axes, including through the center of mass and through the center of the rod.
By understanding these principles, students can better predict and manipulate the rotational behavior of different systems, making this an essential chapter in their physics studies.