Rotational Motion – Stage 1 (Page 6)

Radius of Gyration & Standard Moments of Inertia

After understanding Moment of Inertia, the next important concept is Radius of Gyration. It helps simplify calculations and gives physical insight into mass distribution.

1. Radius of Gyration (k)

Radius of gyration is the distance from the axis of rotation at which the entire mass of the body may be assumed to be concentrated without changing its moment of inertia.

I = M k²

where
I = moment of inertia
M = total mass
k = radius of gyration

2. Physical Meaning of Radius of Gyration

✔ Smaller k → mass concentrated near axis
✔ Larger k → mass spread far from axis
✔ k depends on axis of rotation

Radius of gyration provides a measure of how mass is distributed around the axis.

3. Advantage of Using Radius of Gyration

✔ Simplifies moment of inertia calculation
✔ Avoids summation or integration
✔ Frequently used in IIT/JEE problems

4. Standard Moments of Inertia (Must Memorize)

1. Thin Rod (length L)
• About center, ⟂ to rod: I = ML² / 12
• About one end, ⟂ to rod: I = ML² / 3

2. Circular Ring (radius R)
• About center, ⟂ to plane: I = MR²

3. Solid Disc (radius R)
• About center, ⟂ to plane: I = ½ MR²

4. Solid Sphere
• About diameter: I = 2/5 MR²

5. Hollow Sphere
• About diameter: I = 2/3 MR²

Exam Tip (Very Important)

In exams, you are NOT expected to derive these. You must remember them and apply correctly with axis awareness.

⚠ Common Student Mistakes

❌ Confusing disc with ring ❌ Using wrong axis formula ❌ Forgetting to square radius

IIT-JEE Insight

IIT questions often test recognition of the body and axis. Once identified correctly, the problem becomes straightforward.

Next: Stage 1 – Page 7
Parallel Axis Theorem & Perpendicular Axis Theorem

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