Parallel Axis Theorem

Use the parallel axis theorem: I = I_cm + Md²

where: • I_cm = moment of inertia about center of mass = (1/12)ML² • M = mass of rod • d = distance from center to end = L/2

I = (1/12)ML² + M(L/2)² I = (1/12)ML² + (1/4)ML² I = (1/12)ML² + (3/12)ML² I = (4/12)ML² = (1/3)ML²

The moment of inertia about the end is I = (1/3)ML².

Note: This is larger than I_cm because mass is farther from the axis on average.

For a disk about its center: I_cm = (1/2)MR² I_cm = (1/2)(2 kg)(0.5 m)² = 0.25 kg·m²

Using parallel axis theorem: I = I_cm + Md²

where d = 0.3 m

I = 0.25 + (2)(0.3)² I = 0.25 + 0.18 I = 0.43 kg·m²

The moment of inertia about the parallel axis is 0.43 kg·m².

The pivot is on the surface, so: d = R (distance from center to surface)

Parallel axis theorem: I = I_cm + Md² I = (2/5)MR² + MR² I = (2/5)MR² + (5/5)MR² I = (7/5)MR²

The moment of inertia about the pivot is I = (7/5)MR².

This is why a physical pendulum made from a sphere has different behavior than a simple pendulum - the extended mass distribution matters!

Each rod: I_cm = (1/12)mL² about its center Distance from each rod's center to junction: d = L/2

For each rod using parallel axis theorem: I_rod = I_cm + md² I_rod = (1/12)mL² + m(L/2)² I_rod = (1/12)mL² + (1/4)mL² I_rod = (1/3)mL²

Total for both rods: I_total = 2 × (1/3)mL² = (2/3)mL²

Alternatively, recognizing total mass M = 2m: I_total = (1/3)ML²

The moment of inertia is (2/3)mL² or (1/3)ML².

The edge is distance d = a/2 from the center.

Parallel axis theorem: I = I_cm + Md² I = (1/12)Ma² + M(a/2)² I = (1/12)Ma² + (1/4)Ma²

Convert to common denominator: I = (1/12)Ma² + (3/12)Ma² I = (4/12)Ma² I = (1/3)Ma²

The moment of inertia about the edge is I = (1/3)Ma².

Pattern recognition: For many objects, moving the axis from center to edge increases I by factor of (I_edge/I_cm = 4 for length dimension), following the pattern I = (1/12)ML² → (1/3)ML².