Problem 32
Question
You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. (a) What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod? (b) One of your engineers has proposed to reduce the moment of inertia by bending the rod at its center into a V-shape, with a 60.0\(^\circ\) angle at its vertex. What would be the moment of inertia of this bent rod about an axis perpendicular to the plane of the V at its vertex?
Step-by-Step Solution
Verified Answer
The moment of inertia is 0.012 kg·m² for the straight rod and 0.009 kg·m² for the V-shaped rod.
1Step 1: Understand the Problem
We are given a uniform rod 60.0 cm long with a mass of 0.400 kg. We need to calculate the moment of inertia about an axis at its center and then compare it to when the rod is bent into a V-shape with a 60-degree angle at the vertex.
2Step 2: Calculate the Moment of Inertia for the Straight Rod
The moment of inertia (I) of a rod about an axis through its center and perpendicular to its length is given by the formula: \[ I = \frac{1}{12} m L^2 \] Substituting the given values (where m = 0.400 \text{ kg} and L = 0.60 \text{ m}), we obtain:\[ I = \frac{1}{12} \times 0.400 \times (0.60)^2 = \frac{1}{12} \times 0.400 \times 0.36 \]Calculate the result.
3Step 3: Calculate the Straight Rod Result
Perform the arithmetic:\[ I = 0.012 \text{ kg} \cdot \text{m}^2 \]So, the moment of inertia for the straight rod is 0.012 kg·m².
4Step 4: Understand the V-Shape Configuration
When the rod is bent into a V-shape with a 60° angle at the vertex, we need to consider each half-rod separately since they are symmetrical and form two equal masses joined at the vertex.
5Step 5: Calculate the Moment of Inertia for the V-Shaped Rod
Each leg of the V is 30 cm (half of the original length). Using the parallel axis theorem, for each leg of the V:First, calculate the moment of inertia about the end of a single rod, using:\[ I_{end} = \frac{1}{3} m (\frac{L}{2})^2 = \frac{1}{3} \times 0.200 \times (0.30)^2 \]Then, realize both legs contribute to the total moment of inertia:\[ I = 2 \times I_{end} \cos^2(30^{\circ}) = 2 \times \frac{1}{3} \times 0.200 \times 0.09 \times (\frac{\sqrt{3}}{2})^2 \]Substitute and solve the expression.
6Step 6: Calculate the V-Shaped Rod Result
Perform the arithmetic:\[ I = 2 \times 0.006 \times 0.75 = 0.009 \text{ kg} \cdot \text{m}^2 \]So, the moment of inertia for the V-shaped rod is approximately 0.009 kg·m².
Key Concepts
Uniform RodParallel Axis TheoremV-shape ConfigurationPerpendicular Axis
Uniform Rod
A uniform rod is a slender, elongated solid object where the mass is evenly distributed along its length. This even distribution is crucial because it impacts the rod's physical properties, including its moment of inertia. The moment of inertia \(I\), a measure of how difficult it is to rotate an object about an axis, depends on both the mass of the object and how that mass is distributed with respect to the axis.
For a rod of length \(L\) and mass \(m\), the formula to calculate the moment of inertia about an axis through its center and perpendicular to its length is: \[ I = \frac{1}{12} m L^2 \] This formula considers that a uniform rod rotates more easily than rods with mass concentrated at the ends.
Since the rod in this exercise is uniform, this simplifies the calculation. The rod's length (L = 60 cm or 0.6 m) and its mass (m = 0.4 kg) are plugged into this formula to find its moment of inertia at the specified axis.
For a rod of length \(L\) and mass \(m\), the formula to calculate the moment of inertia about an axis through its center and perpendicular to its length is: \[ I = \frac{1}{12} m L^2 \] This formula considers that a uniform rod rotates more easily than rods with mass concentrated at the ends.
Since the rod in this exercise is uniform, this simplifies the calculation. The rod's length (L = 60 cm or 0.6 m) and its mass (m = 0.4 kg) are plugged into this formula to find its moment of inertia at the specified axis.
Parallel Axis Theorem
The parallel axis theorem is a powerful tool that allows us to find the moment of inertia of an object about any axis, given its moment of inertia about a parallel axis that passes through the object's center of mass.
The theorem states that if you know the moment of inertia about an axis through the center of mass, \(I_{cm}\), you can find \(I\) about any parallel axis a distance \(d\) away by using: \[ I = I_{cm} + m d^2 \] where \(m\) is the mass of the object.
This theorem is particularly useful when analyzing composite configurations like the V-shaped rod. When bending the rod into a V-shape, each half of the rod becomes its separate entity rotationally. We apply the theorem to understand the inertia of the rod when bent, as calculating each half rod's inertia about the vertex and then adjusting for the angle and separation is necessary. This theorem is an essential technique in problems involving combined geometries and shapes that don't naturally align with straightforward axes.
The theorem states that if you know the moment of inertia about an axis through the center of mass, \(I_{cm}\), you can find \(I\) about any parallel axis a distance \(d\) away by using: \[ I = I_{cm} + m d^2 \] where \(m\) is the mass of the object.
This theorem is particularly useful when analyzing composite configurations like the V-shaped rod. When bending the rod into a V-shape, each half of the rod becomes its separate entity rotationally. We apply the theorem to understand the inertia of the rod when bent, as calculating each half rod's inertia about the vertex and then adjusting for the angle and separation is necessary. This theorem is an essential technique in problems involving combined geometries and shapes that don't naturally align with straightforward axes.
V-shape Configuration
When a straight rod is bent into a V-shape, each segment of the rod acts similarly to an individual pendulum or rotating arm pivoting at the vertex. This configuration changes both how the rod distributes its mass relative to the axis and how its moment of inertia is calculated.
In this exercise, when the rod is bent at a 60-degree angle, the rod essentially forms two equal lengths that share a common pivot point at the vertex. Each half rod (30 cm in length) will contribute to the total moment of inertia, but unlike a straight configuration, the contribution now depends on the cosine of the angle due to the angular arrangement: \[ I = 2 \cdot I_{single\ leg} \cdot \cos^2(\theta) \] where \(I_{single\ leg}\) represents the moment of inertia of one half of the rod.
This arrangement results in a more complex calculation but ultimately, as seen in this exercise, yields a reduction in the rod's overall moment of inertia due to its V-shape.
In this exercise, when the rod is bent at a 60-degree angle, the rod essentially forms two equal lengths that share a common pivot point at the vertex. Each half rod (30 cm in length) will contribute to the total moment of inertia, but unlike a straight configuration, the contribution now depends on the cosine of the angle due to the angular arrangement: \[ I = 2 \cdot I_{single\ leg} \cdot \cos^2(\theta) \] where \(I_{single\ leg}\) represents the moment of inertia of one half of the rod.
This arrangement results in a more complex calculation but ultimately, as seen in this exercise, yields a reduction in the rod's overall moment of inertia due to its V-shape.
Perpendicular Axis
The concept of a perpendicular axis is typical in discussions of rotational dynamics for three-dimensional objects. For two-dimensional shapes, the perpendicular axis theorem relates the inertia about an axis perpendicular to the plane to the sum of the inertia about two perpendicular in-plane axes.
In our context with the V-shaped rod, though the perpendicular axis theorem itself doesn’t directly apply, understanding perpendicular axes helps visualize how the inertia calculation changes with the bending of the rod.
By positioning our axis of rotation perpendicular to the plane of the V, and at the vertex, we explore how geometry and mass distribution (in relation to this perpendicular axis) impact the rotational inertia. Such axis selection is key to simplifying problems involving symmetrical shapes like the V-rod configuration and understanding the orientation's impact on dynamics.
In our context with the V-shaped rod, though the perpendicular axis theorem itself doesn’t directly apply, understanding perpendicular axes helps visualize how the inertia calculation changes with the bending of the rod.
By positioning our axis of rotation perpendicular to the plane of the V, and at the vertex, we explore how geometry and mass distribution (in relation to this perpendicular axis) impact the rotational inertia. Such axis selection is key to simplifying problems involving symmetrical shapes like the V-rod configuration and understanding the orientation's impact on dynamics.
Other exercises in this chapter
Problem 30
Small blocks, each with mass \(m\), are clamped at the ends and at the center of a rod of length \(L\) and negligible mass. Compute the moment of inertia of the
View solution Problem 31
A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.300 kg and can be treated as
View solution Problem 35
A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm\(^2\) surrounded by a concentric ri
View solution Problem 36
A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at \(t =\) 0, the wheel turns through 8.20 revolutio
View solution