Problem 79 - University Physics with Modern Physics

Question

A wagon wheel is made entirely of wood. Its components consist of a rim, 12 spokes, and a hub. The rim has mass \(5.2 \mathrm{~kg}\), outer radius \(0.90 \mathrm{~m}\), and inner radius \(0.86 \mathrm{~m}\). The hub is a solid cylinder with mass \(3.4 \mathrm{~kg}\) and radius \(0.12 \mathrm{~m} .\) The spokes are thin rods of mass \(1.1 \mathrm{~kg}\) that extend from the hub to the inner side of the rim. Determine the constant \(c=I / M R^{2}\) for this wagon wheel.

Step-by-Step Solution

Answer

Answer: The constant c for the given wagon wheel is approximately 0.808.

Step 1: Calculate the total mass of the wagon wheel

To find the total mass of the wagon wheel (\(M\)), we will add together the masses of the rim, hub, and spokes: $$ M = m_\text{rim} + m_\text{hub} + 12 \cdot m_\text{spoke} $$ Using the given values, we can calculate the total mass: $$ M = 5.2 \, \text{kg} + 3.4 \, \text{kg} + 12 \cdot 1.1 \, \text{kg} = 5.2 + 3.4 + 13.2 = 21.8 \, \text{kg} $$

Step 2: Calculate the moment of inertia of the rim

To find the moment of inertia of the rim (\(I_\text{rim}\)), it is convenient to consider the rim as a thick ring and use the following formula: $$ I_\text{rim} = \frac{1}{2} \cdot m_\text{rim} \cdot (R_\text{outer}^2 + R_\text{inner}^2) $$ Using the given values for the mass of the rim and its inner and outer radii, we can calculate the moment of inertia of the rim: $$ I_\text{rim} = \frac{1}{2} \cdot 5.2 \, \text{kg} \cdot (0.9^2 + 0.86^2) \, \text{m}^2 = 2.6 (0.81 + 0.7396) = 4.01076 \, \text{kg} \cdot \text{m}^2 $$

Step 3: Calculate the moment of inertia of the hub

The hub is a solid cylinder, so we can use the following formula to calculate its moment of inertia (\(I_\text{hub}\)): $$ I_\text{hub} = \frac{1}{2} \cdot m_\text{hub} \cdot R_\text{hub}^2 $$ Using the given values for the mass and radius of the hub, we can calculate the moment of inertia of the hub: $$ I_\text{hub} = \frac{1}{2} \cdot 3.4 \, \text{kg} \cdot (0.12 \, \text{m})^2 = 1.7 \cdot 0.0144 = 0.02448 \, \text{kg} \cdot \text{m}^2 $$

Step 4: Calculate the moment of inertia of one spoke

We will use the parallel axis theorem to find the moment of inertia of one spoke (\(I_\text{spoke}\)). First, we need to find the moment of inertia of the spoke around an axis through its center of mass, which for a thin rod is: $$ I_\text{spoke, cm} = \frac{1}{12} \cdot m_\text{spoke} \cdot L^2 $$ where \(L\) is the length of the spoke, which is equal to the difference between the outer and inner radii of the rim: $$ L = R_\text{outer} - R_\text{inner} = 0.9 - 0.86 = 0.04 \, \text{m} $$ Now we can calculate the moment of inertia of one spoke around an axis through its center of mass: $$ I_\text{spoke, cm} = \frac{1}{12} \cdot 1.1 \, \text{kg} \cdot (0.04 \, \text{m})^2 = \frac{1}{12} \cdot 1.1 \cdot 0.0016 = 0.00014667 \, \text{kg} \cdot \text{m}^2 $$ Next, we need to use the parallel axis theorem to find the moment of inertia of the spoke around an axis through the hub. According to the theorem: $$ I_\text{spoke} = I_\text{spoke, cm} + m_\text{spoke} \cdot d^2 $$ where \(d\) is the distance from the axis to the center of mass of the spoke: $$ d = \frac{R_\text{outer} + R_\text{inner}}{2} = \frac{0.9 + 0.86}{2} = 0.88 \, \text{m} $$ Now we can calculate the moment of inertia of one spoke around an axis through the hub: $$ I_\text{spoke} = 0.00014667 + 1.1 \cdot 0.88^2 = 0.00014667 + 0.85312 = 0.85327 \, \text{kg} \cdot \text{m}^2 $$

Step 5: Calculate the total moment of inertia of all spokes

There are 12 spokes, so to calculate the total moment of inertia of all spokes (\(I_\text{spokes}\)), we multiply the moment of inertia of one spoke by the number of spokes: $$ I_\text{spokes} = 12 \cdot I_\text{spoke} = 12 \cdot 0.85327 = 10.23924 \, \text{kg} \cdot \text{m}^2 $$

Step 6: Calculate the total moment of inertia of the wagon wheel

To find the total moment of inertia of the wagon wheel (\(I\)), we sum the moments of inertia of the rim, hub, and spokes: $$ I = I_\text{rim} + I_\text{hub} + I_\text{spokes} = 4.01076 + 0.02448 + 10.23924 = 14.27448 \, \text{kg} \cdot \text{m}^2 $$

Step 7: Calculate the constant \(c\)

Finally, we can calculate the constant \(c\) using the formula \(c = \frac{I}{MR^2}\), where \(I\) is the total moment of inertia, \(M\) is the total mass, and \(R\) is the outer radius of the wagon wheel: $$ c = \frac{14.27448}{21.8 \cdot 0.9^2} = \frac{14.27448}{17.646} \approx 0.808 $$ The constant \(c\) for this wagon wheel is approximately \(0.808\).

Key Concepts

Physics Rotational Motion

Understanding rotational motion is essential in physics as it describes the movement of objects spinning around a central axis. This kind of motion is observable in many everyday situations, like wheels turning on a car or the rotation of celestial bodies. To quantify rotational motion, we introduce the concept of angular velocity, which tells us how fast an object rotates, usually measured in radians per second.

Different parts of a rotating object move with different linear speeds, but they share the same angular velocity. A key factor in rotational motion is the distribution of an object's mass relative to its axis of rotation. This leads us to an important property called the moment of inertia (I), which essentially measures the 'rotational inertia' of an object. It depends on both the mass of the object and the distribution of this mass relative to the axis of rotation. The formula for the moment of inertia varies based on the geometry of the object and its mass distribution.

In the example of the wagon wheel, we need to consider the contributions of different components—the rim, the hub, and the spokes. Each component has a unique shape and contributes differently to the wheel's overall moment of inertia because they are at different distances from the axis of rotation. After finding the moments of inertia for each of these components individually, we sum them up to find the total moment of inertia for the wagon wheel in rotational motion.

Parallel Axis Theorem

The parallel axis theorem is a powerful tool used in physics to determine the moment of inertia of an object about any axis, given its moment of inertia about a parallel axis through its center of mass and the perpendicular distance between the two axes. The formula for the parallel axis theorem is
\[ I = I_{\text{cm}} + m d^2 \]
where \( I \) is the moment of inertia about the desired axis, \( I_{\text{cm}} \) is the moment of inertia about the center of mass axis, \( m \) is the mass of the object, and \( d \) is the distance between the two parallel axes.

This theorem is particularly useful for objects like the spokes of the wagon wheel in the provided example. To find the moment of inertia of a single spoke about the wagon wheel’s axis, we first calculate its moment of inertia around its center of mass. Then, we apply the parallel axis theorem, since the spoke’s center of mass is not located on the rotational axis of the wheel. We add the product of the spoke’s mass and the square of the distance from the center of mass to the rotational axis. Multiplying this value by the number of spokes gives us the total contribution of the spokes to the wheel's moment of inertia.

Moment of Inertia of Composite Objects

Many real-world objects, like our example of a wagon wheel, are not made up of a single, homogenous structure but are instead composite objects, consisting of various parts with different shapes, sizes, and materials. Calculating the moment of inertia for such objects involves breaking them down into simpler shapes, calculating the moment of inertia for each shape, and then summing these moments to get the total.

For the wagon wheel, we consider it as a composite of three different shapes: the rim, the spokes, and the hub. Each of these parts has a different moment of inertia formula:

• The rim, treated as a thick ring, has a moment of inertia given by the formula \( I_{\text{rim}} = \frac{1}{2} m (R_{\text{outer}}^2 + R_{\text{inner}}^2) \).
• The hub, as a solid cylinder, has a moment of inertia expressed as \( I_{\text{hub}} = \frac{1}{2} m R^2 \).
• The spokes require a more complex approach using both the formula for a rod about its center of mass and the parallel axis theorem.

By treating each part individually, understanding their unique contributions, and progressively building towards a total, we create a picture of the composite object’s rotational characteristics.

Once all individual moments are calculated, they are added to find the wheel's overall moment of inertia, \( I \). Then, using the formula \( c = \frac{I}{M R^2} \), where \( M \) is the mass of the entire wheel and \( R \) is a relevant characteristic radius (here, the outer radius of the rim), we can compute the constant \( c \) that encapsulates the rotational properties of the wheel. In educational settings, this step-by-step approach illuminates how combining simple principles can address complex systems, a valuable method in both physics problems and broader problem-solving scenarios.