Problem 76 - University Physics with Modern 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 \(\mathrm{V}\) -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 0.00545 kg·m².

Step 1: Understand the Configuration

The rod is bent at its center into a V-shape making each side of the V 25.0 cm long (because the entire rod is 50.0 cm). The angle at the vertex where the two sides meet is 70.0°. We need to find the moment of inertia of this V-shaped object about an axis perpendicular to the plane of the V at its vertex.

Step 2: Calculate The Mass of Each Side

Since the rod is uniform, each side of the V-shaped object will have half the total mass of the rod. Therefore, the mass of each side is 0.320 kg / 2 = 0.160 kg.

Step 3: Determine Moment of Inertia for Each Side Individually

The moment of inertia of a thin rod about an axis through its end and perpendicular to its length is \( \frac{1}{3} m L^2 \), where \(m\) is the mass of the rod, and \(L\) is its length. For one side of the V, \(m = 0.160\, \text{kg}\) and \(L = 0.25\, \text{m}\).Thus, the moment of inertia for one side is \( I_{side} = \frac{1}{3} \times 0.160\, \text{kg} \times (0.25\, \text{m})^2 = 0.00333\, \text{kg} \cdot \text{m}^2\).

Step 4: Adjust for Angle Between Rods Using Perpendicular Axis Theorem

Both legs contribute to the total moment of inertia. Since they meet at an angle other than 180° or 0°, we apply the perpendicular axis theorem:\( I_{total} = 2 \cdot I_{side} \cos(\theta/2) = 2 \cdot 0.00333 \times \cos(35.0°)\)Calculate this as \( cos(35°) \approx 0.8192 \). Hence,\( I_{total} = 2 \cdot 0.00333 \cdot 0.8192 = 0.00545\,\text{kg} \cdot \text{m}^2\).

Step 5: Total Moment of Inertia

Thus, using the symmetry of the configuration, the total moment of inertia of the V-shaped rod about the specified axis is 0.00545 kg·m².

Key Concepts

Perpendicular Axis Theorem

The perpendicular axis theorem is a useful tool in physics, especially when dealing with complex shapes. This theorem relates the moment of inertia of a body about an axis perpendicular to its plane, with its moments of inertia about two perpendicular axes in the plane of the body. To be more specific, if you have a flat shape laying in the x-y plane, the theorem states that the moment of inertia about the z-axis (perpendicular to the plane) is the sum of its moments of inertia about the x and y axes. Mathematically:

• \( I_z = I_x + I_y \)

This principle is often applied to objects with symmetry, where calculating inertia directly might be complex. In the case of our V-shaped rod, although not directly in the plane, the two rods are laid out symmetrically such that projections can be made using the perpendicular axis theorem to understand their combined effect.

V-shaped Rod

A V-shaped rod introduces interesting dynamics due to its shape and angle. When a rod is bent into a V-shape, each limb makes a distinct contribution to its overall moment of inertia. Each leg of the V has the same length and mass because the rod was initially straight and uniform. Upon bending, the angle at the vertex becomes crucial as it influences how the moments of inertia of each leg interact when calculating for axes not aligned with each individual leg. The angle determines how much each side resists rotational motion. This angle, which can be difficult to capture without the right mathematical tool (like trigonometry), impacts both the magnitude and the symmetry of the inertia distribution.

Uniform Rod

Uniform rods are rods that have a consistent mass distribution throughout their length. This uniformity simplifies calculations because each section of the rod contributes equally to its moment of inertia. For linear rods, the center of mass and other properties are predictable and calculations can often rely on symmetry. In the exercise, we are dealing with a rod of uniform mass and length, making it straightforward to determine that each side of the V-shaped rod has half of the total mass. Uniform rods are crucial because they lay down a basis for predictability in problems concerning physical objects without varying density or geometry.

Angular Mechanics

Angular mechanics deals with objects in rotation, studying the effects of forces that cause such motion. It goes beyond linear mechanics by considering quantities like torque, angular momentum, and the moment of inertia. The moment of inertia reflects how mass is distributed relative to the rotational axis. This concept is vital as it dictates how an object will react to applied rotational forces. In angular mechanics, studying scenarios like the V-shaped rod allows us to understand how non-standard shapes impact motion. Such a shape's collective resistance to change, or its rotational inertia, depends heavily on its geometry and mass distribution, both of which angular mechanics helps in understanding comprehensively.