Question

At the grocery store you pick up a can of beef broth and a can of chunky beef stew. The cans are identical in diameter and weight. Rolling both of them down the aisle with the same initial speed, you notice that the can of chunky stew rolls much farther than the can of broth. Why?

Short answer

The can of chunky stew rolls farther because it likely has a lower rotational inertia compared to the broth.

Step-by-step solution

Understanding the Problem

We need to determine why the can of chunky beef stew rolls farther than the can of beef broth, given they have the same diameter and weight.

Analyzing the Composition of the Cans

The can of chunky beef stew likely has a different mass distribution compared to the can of broth. The liquid broth has a more uniform mass distribution, while the chunky stew likely has a non-uniform distribution with some mass concentrated at different radial distances from the center.

Exploring Rotational Inertia

Rotational inertia, or moment of inertia, depends on how mass is distributed with respect to the axis of rotation. The can of broth, with uniform mass distribution, probably has a higher rotational inertia compared to the chunky stew, which might have more mass closer to the center.

Applying Rotational Dynamics

A can with lower rotational inertia (chunky stew) is easier to accelerate rotationally than a can with higher rotational inertia (broth). This means it will lose less energy to rotational motion and maintain its translational speed longer, rolling farther.

Key concepts

Moment of Inertia

The moment of inertia, often referred to as rotational inertia, is a fundamental concept in physics that describes how mass is distributed in relation to an axis of rotation. Essentially, it determines how easy or difficult it is to change an object's rotational motion. The greater the moment of inertia, the more difficult it is to start or stop the object’s rotation. It is represented mathematically as:\[I = \sum m_i r_i^2\]where \( I \) is the moment of inertia, \( m_i \) is the mass of the individual particle, and \( r_i \) is the perpendicular distance from the axis of rotation to the particle. This formula shows that the mass and how far it is from the axis contribute greatly to the moment of inertia. In the context of the two cans at the grocery store, the can of broth, with its likely uniform mass, will have particles distributed across the can’s volume, often making it harder to rotate due to a larger moment of inertia.

Mass Distribution

Mass distribution affects rotational motion significantly by determining how mass is spread out relative to the axis of rotation. The way mass is distributed inside an object influences its moment of inertia. For objects like the can of chunky beef stew and the can of broth, the difference lies primarily in their internal composition.

• Uniform Mass Distribution: The broth inside the can likely spreads evenly, making the mass distribution uniform. This is typical in liquids, where mass is spread out evenly leading to a larger moment of inertia.
• Non-uniform Mass Distribution: Chunky stew consists of irregular chunks distributed variably. Some mass may be closer to the center, reducing the overall moment of inertia, making it easier to rotate.

This variance in mass distribution between the two cans accounts for their differing rotational capabilities and how far they roll when set into motion.

Rotational Dynamics

Rotational dynamics is the study of objects that are rotating or spinning. When considering why the chunky stew travels farther than the broth, we turn to rotational dynamics to understand the forces at play. A key principle here is the relationship between rotational inertia and rotational motion:

• Energy Conservation: In a rolling motion scenario, some energy goes into translational motion and some into rotational motion. An object with lower rotational inertia, like the chunky stew, reserves more energy for translating down the aisle.
• Acceleration and Motion: A lower moment of inertia means the object can pick up rotational speed more easily. The can of chunky stew, therefore, uses less energy to spin and can maintain its initial speed better than the broth.

Overall, rotational dynamics illustrates why the same-force applied scenario results in the chunky stew rolling a greater distance. The distribution of mass influences how much energy is used for rotation versus translation, dictating the final distances reached by the cans.