Problem 34 - College Physics

Question

A wagon wheel is constructed as shown in Figure \(9.31 .\) The radius of the wheel is \(0.300 \mathrm{~m}\), and the rim has a mass of \(1.40 \mathrm{~kg}\). Each of the wheel's eight spokes, which come out from the center and are 0.300 m long, has a mass of \(0.280 \mathrm{~kg}\). What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel?

Step-by-Step Solution

Answer

0.1932 kg·m^2

Step 1: Identify components

To find the moment of inertia of the wheel, identify and separate the components that contribute to it. There are two primary parts: the rim and the eight spokes.

Step 2: Calculate rim's moment of inertia

The rim can be treated as a hoop with mass concentrated along the circumference. The moment of inertia for a hoop is \[ I_{ ext{rim}} = m_{ ext{rim}} r^2 \]where \( m_{\text{rim}} = 1.40 \, \text{kg} \) and \( r = 0.300 \, \text{m} \). Thus: \[ I_{\text{rim}} = 1.40 \times (0.300)^2 = 0.126 \text{ kg} \cdot \text{m}^2 \]

Step 3: Calculate spoke's moment of inertia

Each spoke can be treated as a thin rod rotating about one end. The moment of inertia for a single spoke is \[ I_{ ext{spoke}} = \frac{1}{3} m_{ ext{spoke}} L^2 \]where \( m_{\text{spoke}} = 0.280 \, \text{kg} \) and \( L = 0.300 \, \text{m} \). Calculate moment of inertia for one spoke: \[ I_{\text{spoke}} = \frac{1}{3} \times 0.280 \times (0.300)^2 = 0.0084 \text{ kg} \cdot \text{m}^2 \]

Step 4: Calculate moment of inertia for all spokes

Since there are eight identical spokes, the total moment of inertia for all spokes is: \[ I_{ ext{total ext{ }spokes}} = 8 \times I_{ ext{spoke}} = 8 \times 0.0084 = 0.0672 \text{ kg} \cdot \text{m}^2 \]

Step 5: Combine moment of inertia for rim and spokes

To get the total moment of inertia of the wheel, sum the moments of inertia of the rim and all spokes: \[ I_{ ext{total}} = I_{ ext{rim}} + I_{ ext{total ext{ }spokes}} = 0.126 + 0.0672 = 0.1932 \text{ kg} \cdot \text{m}^2 \]

Step 6: Conclusion

The total moment of inertia of the wheel, considering the rim and spokes, is \( 0.1932 \text{ kg} \cdot \text{m}^2 \).

Key Concepts

wheel dynamics

In understanding wheel dynamics, it is crucial to consider all parts of the wheel. A wheel typically consists of a rim and spokes, and these components contribute to its moment of inertia. Here’s how it works:

• The rim is the outermost part of the wheel, often treated as a hoop.
• The spokes are the components that radiate from the center, like thin rods.

Each part contributes to the wheel's dynamics, influencing how it rotates and how its mass is distributed. Analyzing these elements helps in calculating the moment of inertia effectively, which, in turn, describes how difficult it is to change the wheel's rotational state.

rotational motion

Rotational motion involves objects spinning around an axis. Like linear motion, it has parameters such as velocity and acceleration, but they take on angular forms (e.g., angular velocity, angular acceleration). In the case of the wagon wheel:

• The rim and spokes spin about the central axis.
• Each point on the wheel undergoes circular motion, characterized by its radius.

The moment of inertia is essential here as it dictates how the mass of the wheel affects its rotational energy and resistance to changes in motion. It's akin to mass in linear dynamics, where it affects acceleration when a force is applied.

physics problem-solving

Physics problem-solving often involves breaking down complex systems into simpler parts. For the wagon wheel, this means analyzing the rim and spokes separately to determine their individual moments of inertia. The process typically follows these steps:

• Identify each component of the system (e.g., rim, spokes).
• Calculate the individual contribution of each part to the total moment of inertia.
• Combine the results to find the overall effect.

This methodical approach helps manage complexity and ensures accuracy, applying fundamental physics principles like superposition in calculations.

mechanical physics

Mechanical physics encompasses the study of motion and the forces that produce it. With reference to the wagon wheel, the study of mechanical physics provides insights into how forces like friction and tension interact with rotational aspects such as torque. In the realm of rotation:

• Torque is the force causing the wheel to begin spinning.
• Friction often acts against motion, affecting how readily a wheel begins or continues to rotate.

Understanding these principles enables us to predict and control the motion effectively. The moment of inertia becomes a crucial factor, as it influences how much torque is needed to achieve desired accelerations in the wheel.