MOI of Combined Bodies Trick 🔥 | Parallel Axis

 

❓ Concept Question

How do we calculate the moment of inertia of a system of different bodies about a given axis?

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🖼 Concept Image

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✍️ Short Concept

MOI depends on:

👉 Mass distribution
👉 Shape of body
👉 Axis of rotation

For multiple bodies → calculate individually and then add.

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🔷 Step 1 — MOI Depends on Axis 💯

Moment of inertia is not fixed.

It depends on:

• Shape
• Mass distribution
• Axis of rotation

👉 Axis change ⇒ MOI change

This is the most important idea.

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🔷 Step 2 — Different Bodies, Different MOI

Even if mass and radius same, formulas differ:

Disc → $\frac{1}{4}MR^2$ (about diameter)

Solid sphere → $\frac{2}{5}MR^2$

Spherical shell → $\frac{2}{3}MR^2$

📌 JEE mixes these to create confusion.

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🔷 Step 3 — Reference MOI Trick

Often given:

$$I = \text{MOI of disc about its diameter}$$

👉 Convert all other MOIs into multiples of I

This simplifies calculation massively.

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🔷 Step 4 — Parallel Axis Theorem (MOST IMPORTANT 🚨)

If axis is shifted:

$$I = I_{cm} + Md^2$$

Where:

d = distance between axes

👉 Apply this separately for each body.

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🔷 Step 5 — Total MOI of System

Total MOI:

$$I_{net} = I_1 + I_2 + I_3$$

Add contributions of all bodies.

⚠️ Be careful about:

• Distances
• Axis orientation
• Correct formula

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✅ Final Takeaway

For combined systems:

$$\boxed{I_{net} = \sum (I_{cm} + Md^2)}$$

Final answer often comes as:

$$\frac{x}{\text{constant}} \cdot I$$

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⭐ Golden JEE Insight

MOI increases when:

👉 Mass moves away from axis

So always check:

Distance ↑ ⇒ MOI ↑