Problem 67 - College Physics

Question

A thin uniform rod 50.0 \(\mathrm{cm}\) long with mass 0.320 \(\mathrm{kg}\) is bent at its center into a \(\mathrm{V}\) shape, with a \(70.0^{\circ}\) angle at its vertex. Find the moment of inertia of this V-shaped object about an axis perpendicular to the plane of the \(\mathrm{V}\) at its vertex.

Step-by-Step Solution

Answer

The moment of inertia is approximately \(0.0149 \, \text{kg} \, \text{m}^2\).

Step 1: Identify Moment of Inertia Formula

The moment of inertia \( I \) for a rod of length \( L \) and mass \( m \) about an axis through its center is given by the formula \( I = \frac{1}{12} m L^2 \). In this case, the rod is divided into two equal parts of \( 25 \text{ cm} \) each.

Step 2: Calculate Mass of Each Rod Segment

Since the rod is uniform and bent at its center, each segment is \( 25 \text{ cm} \). Hence, the mass of each segment is \( \frac{0.320}{2} \ \mathrm{kg} = 0.160 \ \text{kg} \).

Step 3: Moment of Inertia of Each Segment About Its Own Center

For each segment, the moment of inertia about its own center: \( I = \frac{1}{12} m L^2 = \frac{1}{12} \times 0.160 \times (0.25)^2 \). This calculates to approximately \( 0.000833 \ \text{kg} \, \text{m}^2 \) for each segment.

Step 4: Apply the Parallel Axis Theorem

The Parallel Axis Theorem states \( I = I_{cm} + md^2 \) where \( d \) is the perpendicular distance from the centroid to the axis. Here, \( d = \frac{L}{2}\cos(\theta / 2) \). So \( d = 0.25 \times \cos(35^\circ) \approx 0.204 \ \text{m} \).

Step 5: Calculate Each Segment's Inertia About Vertex Axis

Inertia of each segment about the vertex using the parallel axis theorem: \( I_{segment} = I + md^2 = 0.000833 + 0.160 \times 0.204^2 \). This results in approximately \( 0.007467 \ \text{kg} \, \text{m}^2 \) per segment.

Step 6: Total Moment of Inertia for the V-shape

Since there are two such segments, the total moment of inertia \( I_{total} = 2 \times 0.007467 \ \text{kg} \, \text{m}^2 = 0.014934 \ \text{kg} \, \text{m}^2 \).

Key Concepts

Parallel Axis Theorem

The Parallel Axis Theorem is a pivotal concept in rotational dynamics. It helps us find the moment of inertia of an object around any axis, provided we know the moment of inertia around a parallel axis that passes through the object's center of mass (COM). Think of it as a handy tool for shifting the axis of rotation without recalculating from scratch.

The theorem mathematically states that the moment of inertia about any axis is the sum of its moment of inertia about a parallel axis through the center of mass and the product of its mass and the square of the perpendicular distance between the two axes. In formula terms, this 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 of the object,
• \( d \) is the distance between the two parallel axes.

Utilizing this theorem, you can calculate the inertia for more complex shapes like the V-shaped rod by treating each segment separately and then combining their results.

Centre of Mass

The centre of mass (COM) is the specific point where the entire mass of an object is imagined to be concentrated for calculations pertaining to translational and rotational motion.

For a uniformly distributed object such as a rod, its center of mass is typically in the middle. When a rod is bent, things become a tad more complex. Each section might have its own center of mass when bent, making it essential to analyze each segment separately, as they affect how the object rotates.

• In this V-shaped rod scenario, the COM for rotational axis considerations intrigues us more than the COM of the entire object, as we consider rotational dynamics about the vertex.

Positioning your axis at the COM simplifies calculations and is crucial in applying concepts like the Parallel Axis Theorem effectively.

Rotational Dynamics

Rotational dynamics, distinct from linear dynamics, examines how objects spin or rotate. Just like linear dynamics relies heavily on concepts of mass and acceleration, rotational dynamics pivots on moment of inertia and angular acceleration.

To understand rotational dynamics, appreciate how moment of inertia plays the role that mass does in linear motion—it is a measure of an object's resistance to change in its rotational motion.

• A higher moment of inertia means more effort is required to change the object's spinning speed.
• It's not just about mass—how mass is distributed relative to the pivot point matters too.
In rotating systems, angular velocity, angular acceleration, and torque work collaboratively, akin to velocity, acceleration, and force in linear systems.
• Equations like \( \tau = I \alpha \) (torque equals moment of inertia times angular acceleration) play a key role.