Difficulty: Easy
Correct Answer: I = (1/2) * m * r^2
Explanation:
Introduction / Context: The mass moment of inertia quantifies resistance to angular acceleration about a specific axis. For a solid cylinder, the polar (longitudinal) axis is the central axis of symmetry; knowing I for this axis is essential in rotor dynamics and torsional vibration problems.
Given Data / Assumptions:
Concept / Approach: For bodies of revolution, standard results give compact expressions for I. The solid cylinder about its own axis has a classical closed form derived from integrating r^2 over the cross-sectional disk and extruding along length l.
Step-by-Step Solution:
Definition: I = ∫ r^2 dm over the volume. For a disk element of thickness dz, I_disk = (1/2) * ρ * π * r^4 per unit thickness; integrating along l multiplies by l and relates to total mass m. Result simplifies to I = (1/2) * m * r^2.Verification / Alternative check: Compare with a thin ring of radius r: I_ring = m * r^2. A solid cylinder concentrates more mass near the centre than a ring, so I must be smaller; (1/2) m r^2 is consistent physically.
Why Other Options Are Wrong: Fractions of 1/4, 1/6, 1/8 do not match the standard integral for a solid cylinder about its axis and yield values that are too small compared with established results.
Common Pitfalls: Confusing polar axis (along the cylinder) with diametral axis (across the cylinder), which has a different I value: I_diametral = (1/4) m r^2 + (1/12) m l^2.
Final Answer: I = (1/2) * m * r^2.
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