Problem 31 - College Physics

Question

Four small \(0.200 \mathrm{~kg}\) spheres, each of which you can regard as a point mass, are arranged in a square \(0.400 \mathrm{~m}\) on a side and connected by light rods. (See Figure \(9.29 .)\) Find the moment of inertia of the system about an axis (a) through the center of the square, perpendicular to its plane at point \(O ;\) (b) along the line \(A B ;\) and (c) along the line \(C D\).

Step-by-Step Solution

Answer

(a) 0.064 kg m²; (b) 0.064 kg m²; (c) 0.064 kg m².

Step 1: Understanding the Problem

We are asked to find the moment of inertia of a system of four point masses arranged in a square. Each sphere has a mass of \(0.200 \text{ kg}\). The square has a side length of \(0.400 \text{ m}\). We will calculate the moment of inertia about three different axes.

Step 1: Moment of Inertia through the Center (Axis through O)

To find the moment of inertia through the center and perpendicular to the plane of the square, apply the formula \( I = \sum m_i r_i^2 \), where \( r_i \) is the distance from each mass to the rotational axis.Each sphere is along the diagonal from the center, so \( r = \frac{\sqrt{2}}{2} \times 0.400 \). The distance \( r = 0.283 \text{ m} \).Thus, the moment of inertia is:\[I = 4 \times 0.200 \times (0.283)^2 = 4 \times 0.200 \times 0.08 = 0.064 \text{ kg m}^2\]

Step 2: Moment of Inertia along the Line AB

For the axis along the line AB, only the spheres at points C and D contribute since the distance for A and B is zero. For C and D:\( r = 0.400 \text{ m} \).Hence,\[I = 2 \times 0.200 \times (0.400)^2 = 2 \times 0.200 \times 0.16 = 0.064 \text{ kg m}^2\]

Step 3: Moment of Inertia along the Line CD

For an axis along line CD, only the spheres at points A and B will contribute to the moment of inertia, since C and D are on the axis. Therefore:\( r = 0.400 \text{ m} \).Therefore, the moment of inertia is:\[I = 2 \times 0.200 \times (0.400)^2 = 2 \times 0.200 \times 0.16 = 0.064 \text{ kg m}^2\]

Key Concepts

Point Mass and Its Role in Physics

In physics, the concept of a point mass is an idealization where the mass of an object is assumed to be concentrated at a single point. This simplification is especially useful when analyzing the motion of objects, allowing for straightforward calculations of physical quantities like momentum and energy. In the case of the exercise involving the four small spheres, each sphere is considered a point mass. This approach helps streamline the calculations for the moment of inertia— a crucial factor in understanding rotational dynamics.
The benefit of treating objects as point masses simplifies the mathematics behind many physical phenomena. It permits the use of simple algebra rather than complex integrals, making it accessible for educational purposes. When objects' specific distributions don't significantly affect the overall calculations (due to symmetry or distribution), considering them as point masses is often a preferred approach.

Exploring Rotational Axis

A rotational axis is an imaginary line around which an object rotates. It acts like a pivot point, helping us understand how the distribution of mass affects an object's rotational motion. In the exercise, we evaluate the moment of inertia about three different axes: through the center of the square (at point O), along the line AB, and along the line CD.
The axis through the center allows us to see how each mass is distributed with respect to the center. When an axis goes through two points on an object, like AB or CD, only parts of the object not lying directly along the axis contribute to the rotation. This distinction is essential. The axis' placement significantly affects the calculations, and thus, the understanding of the object's rotational inertia.

The Step-by-Step Physics Problem Solving Approach

Solving physics problems step by step is an essential approach in learning and understanding concepts effectively. It involves breaking down a complex problem into smaller, more manageable parts. The solution to the exercise on the moment of inertia of spheres involves such an approach.
First, understanding the basics, like identifying the masses and distances involved, sets the groundwork. Subsequent steps utilize formulas tailor-fitted to the specific scenarios presented by different axes. For instance, using the formula \[ I = \sum m_i r_i^2 \] for calculating the moment of inertia ensures that all contributing factors are accounted for. This method not only aids in deriving the correct solution but enhances the learner's ability to tackle similar problems by developing a consistent workflow.

Understanding Square Arrangement in Rotational Problems

A square arrangement, as described in the exercise, provides a symmetric setup for analyzing rotational motion. Symmetry in such problems means calculations can be greatly simplified. In a square, all sides are equal, and diagonal elements are symmetrical around the center.
When calculating moment of inertia, this symmetry allows for consistent and predictable distances from masses to any chosen axis. This is particularly evident in configurations like the one presented: four point masses at equal positions along a square's corners. Such regularity aids in simplifying the problem, reducing potential sources of error and helps illustrate the principles of rotational dynamics through clear, calculable outcomes.

• All masses equidistant from an axis through the center influence it similarly.
• On sides like AB and CD, symmetry limits contributing masses, simplifying calculations.

Understanding this arrangement aids greatly when it comes to visualizing and calculating rotational dynamics.