Question

A wagon wheel is made entirely of wood. Its components consist of a rim, 12 spokes, and a hub. The rim has mass \(5.20 \mathrm{~kg}\), outer radius \(0.900 \mathrm{~m}\), and inner radius \(0.860 \mathrm{~m}\). The hub is a solid cylinder with mass \(3.40 \mathrm{~kg}\) and radius \(0.120 \mathrm{~m}\). The spokes are thin rods of mass \(1.10 \mathrm{~kg}\) that extend from the hub to the inner side of the rim. Determine the constant \(c=I / M R^{2}\) for this wagon wheel.

Short answer

Based on the given properties of the wagon wheel, determine the constant "c" which is defined as the ratio of the moment of inertia to the product of the mass and the square of the radius. Solution: The constant "c" for this wagon wheel is approximately 0.744.

Step-by-step solution

Calculate the total mass of the wheel

The total mass of the wheel is the sum of the masses of the rim, spokes, and the hub. $$M_{total} = M_{rim} + M_{spokes} + M_{hub}$$ $$M_{total} = 5.20 \mathrm{~kg} + 12(1.10 \mathrm{~kg}) + 3.40 \mathrm{~kg}$$ $$M_{total} = 18.60 \mathrm{~kg}$$

Calculate the moment of inertia for each component

For the rim, the moment of inertia can be calculated using the following formula for a thin ring: $$I_{rim} = \frac{1}{2}M_{rim}(R_{outer}^{2} + R_{inner}^{2})$$ $$I_{rim} = \frac{1}{2}(5.20 \mathrm{~kg})(0.900^2 + 0.860^2)$$ $$I_{rim} = 3.87156 \mathrm{~kg\cdot m^2}$$ For the hub, the moment of inertia can be calculated using the following formula for a solid cylinder: $$I_{hub} = \frac{1}{2}M_{hub}R_{hub}^{2}$$ $$I_{hub} = \frac{1}{2}(3.40 \mathrm{~kg})(0.120^2)$$ $$I_{hub} = 0.02448 \mathrm{~kg\cdot m^2}$$ For the spokes, the moment of inertia for each individual spoke can be calculated using the parallel axis theorem, and then multiplied by the number of spokes to get the total moment of inertia for all spokes: $$I_{spoke} = \frac{1}{3}M_{spoke}L^{2} + M_{spoke}R_{inner}^{2}$$ $$I_{spokes} = 12 \times I_{spoke}$$ $$I_{spokes} = 12 \times ((\frac{1}{3})(1.10 \mathrm{~kg})(0.860)^2 + (1.10 \mathrm{~kg})(0.860)^2)$$ $$I_{spokes} = 7.633184 \mathrm{~kg\cdot m^2}$$

Calculate the total moment of inertia

Now, let's find the total moment of inertia by summing up the moment of inertia for each of the components: $$I_{total} = I_{rim} + I_{spokes} + I_{hub}$$ $$I_{total} = 3.87156 \mathrm{~kg\cdot m^2} + 7.633184 \mathrm{~kg\cdot m^2} + 0.02448 \mathrm{~kg\cdot m^2}$$ $$I_{total} = 11.529224 \mathrm{~kg\cdot m^2}$$

Calculate the constant "c"

Finally, we can find the constant "c" by dividing the total moment of inertia by the product of the total mass and radius squared: $$c = \frac{I_{total}}{M_{total}R_{outer}^{2}}$$ $$c = \frac{11.529224 \mathrm{~kg\cdot m^2}}{(18.60 \mathrm{~kg})(0.900^2)}$$ $$c = 0.744386$$ The constant "c" for this wagon wheel is approximately 0.744.

Key concepts

Moment of Inertia

Moment of inertia is a physical quantity that represents how difficult it is to change the rotational motion of an object. It plays a pivotal role in rotational dynamics as it is the rotational equivalent of mass for linear motion. The moment of inertia depends not only on the mass of an object but also on how that mass is distributed with respect to the axis of rotation. The further the mass is from the axis, the larger the moment of inertia.

Different shapes have different moments of inertia formulas. For instance, for a solid cylinder rotating about its center, the moment of inertia is given by \( I = \frac{1}{2}MR^2 \), where \( M \) is the mass and \( R \) is the radius. The calculation becomes more complex when dealing with composite objects, as the individual moments of inertia of each part must be considered and summed to find the total moment of inertia.

Physics Problem Solving

In physics problem solving, it's important to systematically break down a complex problem into smaller, more manageable parts. This approach allows for a step-by-step solution, ensuring that each aspect of the problem is addressed. In our wagon wheel example:

• Identify the system: Analyze the components - the rim, spokes, and hub.
• Understand the theory: Use the appropriate physics concepts, such as the formulae for moments of inertia for different shapes.
• Perform calculations: Calculate the moment of inertia for each component individually before summing them up.
• Sanity check: Review each step to ensure no part of the problem has been overlooked or miscalculated.

This methodical approach forms the cornerstone of effective physics problem solving.

Rotational Dynamics

Rotational dynamics concerns the behavior of rotating systems and includes the analysis of rotational motion using concepts like torque, angular momentum, and moment of inertia. Understanding how these quantities interrelate is essential for solving rotational motion problems.

In terms of energy, rotating objects possess kinetic energy just like objects in translational motion. The kinetic energy of a rotating body is \( K = \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. By accounting for all forces and torques involved, we can predict and describe the rotational behavior of objects such as the aforementioned wagon wheel.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass, provided we know the moment of inertia about the center of mass axis. The theorem is expressed as \( I = I_{CM} + Md^2 \), where \( I \) is the moment of inertia about the new axis, \( I_{CM} \) is the moment of inertia about the center of mass axis, \( M \) is the total mass, and \( d \) is the distance between the two axes.

Applying the Parallel Axis Theorem

In the wagon wheel problem, the parallel axis theorem is crucial for finding the moment of inertia of the spokes because they rotate around the wheel's hub, not their own centers. By using the theorem, we can account for the additional rotational inertia due to this displacement, allowing us to accurately calculate the wheel's total moment of inertia.