Area Moments — Solid Cylinder about Longitudinal (Polar) Axis Find the mass moment of inertia I of a solid circular cylinder (mass m, radius r, length l) about its longitudinal/polar axis (through the centre, along the axis).

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:

• Solid cylinder: uniform density, radius r, length l, mass m.
• Axis of rotation: cylinder’s own axis (polar/longitudinal).

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:

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.