This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Here's a question and its solution on the moment of inertia for an object made of two concentric circles: --- Question: A uniform, flat disk has an outer radius R = 0.50 m and a concentric circular hole with an inner radius r = 0.20 m. The total mass of the remaining annular (ring-shaped) object is M = 12 kg. Calculate the moment of inertia of this annular object about an axis perpendicular to its plane and passing through its center. --- Solution: Step 1: Identify the formula for the moment of inertia of an annulus (a disk with a concentric hole). The moment of inertia (I) for an annulus with total mass M, outer radius R, and inner radius r, about an axis perpendicular to its plane and passing through its center, is given by: I = (1/2)M(R² + r²) Step 2: List the given values. Mass, M = 12 kg Outer radius, R = 0.50 m Inner radius, r = 0.20 m Step 3: Substitute the values into the formula. I = (1/2) × 12 kg × ((0.50 m)² + (0.20 m)²) Step 4: Calculate the squares of the radii. R² = (0.50 m)² = 0.25 m² r² = (0.20 m)² = 0.04 m² Step 5: Add the squared radii. R² + r² = 0.25 m² + 0.04 m² = 0.29 m² Step 6: Perform the final multiplication. I = (1/2) × 12 kg × 0.29 m² I = 6 kg × 0.29 m² I = 1.74 kg·m² The moment of inertia of the annular object is 1.74 kg·m². --- Send me the next one 📸