Problem 73 - College Physics

Question

A new species of eel is found to have the same mass but one-quarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has A. half the moment of inertia of the American eel. B. the same moment of inertia as the American eel. C. twice the moment of inertia of the American eel. D. four times the moment of inertia of the American eel.

Step-by-Step Solution

Answer

The new eel has four times the moment of inertia of the American eel (Answer: D).

Step 1: Understanding the Problem

To determine how the moment of inertia of a new species of eel compares to that of the American eel, we first need to consider the formula for the moment of inertia for a cylindrical object like an eel rotating around its long axis. The formula is \( I = \frac{1}{2} m r^2 \), where \( m \) is mass and \( r \) is the radius (half the diameter).

Step 2: Analyzing the New Eel's Dimensions

The new eel has the same mass as the American eel but has one-quarter the length and twice the diameter. Since the diameter is twice, the radius \( r_{new} \) is also twice that of the American eel, so \( r_{new} = 2 \times r_{original} \).

Step 3: Calculating Moment of Inertia Relations

Insert the radius relation into the moment of inertia formula: \( I_{new} = \frac{1}{2} m (2r_{original})^2 \). Simplifying gives \( I_{new} = \frac{1}{2} m (4r_{original}^2) = 4 \times \frac{1}{2} m r_{original}^2 = 4I_{original} \).

Step 4: Comparing New and Original Moment of Inertia

Thus, the moment of inertia of the new species of eel is four times that of the American eel. Therefore, the correct answer is: D. four times the moment of inertia of the American eel.

Key Concepts

Cylindrical Object

When we think about the moment of inertia of an object, visualizing the shape can help us understand its rotational dynamics. In this exercise, both the American eel and the new eel species are considered as cylindrical objects. A cylinder is a three-dimensional shape with two parallel bases that are circular and an elongated body, which in this case, represents the body of the eels.

For objects like these, when they rotate around their long axis (like spinning around their length), their dimensions heavily influence the moment of inertia. The moment of inertia for a cylinder rotating about its long axis is given by the formula \( I = \frac{1}{2} m r^2 \). Here, \( m \) is the mass of the cylinder, which remains constant in this problem, and \( r \) is the radius, or half the diameter, which relates to the diameter measurements of the eel. Understanding this formula helps us see how changes in geometry affect rotational properties.

In physics, simplifying complex shapes into basic geometrical ones, like a cylinder, helps in easily calculating physical properties like the moment of inertia. This concept is key in many physics problems involving rotation.

American Eel

The American eel is an intriguing animal often studied in various scientific fields, not just biology but physics as well. It serves as the perfect subject for our moment of inertia problem because of its cylindrical shape.

The American eel is characterized by its elongated, snake-like body. In this exercise, we know that its mass remains constant, even when compared to a different species with modified dimensions. What's fascinating here is how its physical parameters—length and diameter—get adjusted without changing its mass, serving as a reminder of how nature's design can present intriguing problems in physics.

By choosing the American eel for this exercise, we emphasize the interplay of biological structure and physical principles. This gives students practical insight into how theoretical physics principles apply to living creatures, bridging the gap between life sciences and physical sciences.

Rotational Dynamics

Rotational dynamics focuses on the motion of objects that rotate. In this context, we examine how the cylindrical shape and dimensions of the eel impact its rotational motion.

One essential concept in rotational dynamics is the moment of inertia, which is analogous to mass in linear motion. The moment of inertia quantifies how difficult it is to change an object's rotational state. For a cylindrical object like an eel, we use the formula \( I = \frac{1}{2} m r^2 \), where the ease of rotation changes according to the radius.

In this exercise, when the radius of the new eel species is doubled while keeping the mass constant, the complexity of rotational dynamics becomes evident. Squaring the radius results in a fourfold increase in the moment of inertia, revealing how sensitive it is to changes in dimensions. Such problems highlight the intricate relationship between geometry and rotational behavior, providing an intriguing challenge in physics studies.

Physics Problem Solving

Solving physics problems requires a blend of understanding fundamental concepts and applying logical approaches. In this instance, the task involves comparing the moments of inertia between two eels with different dimensions but identical masses.

To tackle this problem:

• First, identify the relevant physical characteristics and how they relate to formulas. Recognize that geometry greatly influences moment of inertia.
• Next, correctly apply the moment of inertia formula \( I = \frac{1}{2} m r^2 \) to derive the relationships needed.
• Analyzing each species' dimensions and translating these into the formula reveals how changes affect the moment of inertia.
• Finally, interpret the results logically, justifying each step to ensure comprehensive understanding.

By breaking down the steps and focusing on precise execution, students can efficiently solve complex physics problems, gaining confidence in their abilities to tackle real-world and theoretical challenges alike.