MOI of Combined Bodies Using Parallel Axis

 

❓ Question

A, B and C are:

• Disc

• Solid sphere

• Spherical shell

All have same mass (M) and radius (R).

Moment of inertia of system about PQ axis is:

$$\frac{x}{15} I$$

Where:

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

Find x.

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🖼 Question Image

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✍️ Short Concept

This is a standard MOI formula + addition question.

Golden idea:

👉 Write MOI of each body about its own center
👉 Convert to required axis (if needed)
👉 Add them
👉 Compare with given I

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🔷 Step 1 — Write Standard MOI Formulas 💯

For same mass M and radius R:

🔹 Disc (about diameter)

$$I_d = \frac{1}{4} MR^2$$

(This is given as I)

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🔹 Solid Sphere (about diameter)

$$I_s = \frac{2}{5} MR^2$$

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🔹 Spherical Shell (about diameter)

$$I_{sh} = \frac{2}{3} MR^2$$

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🔷 Step 2 — Express Everything in Terms of I

Given:

$$I = \frac{1}{4} MR^2$$

So,

Solid sphere:

$$\frac{2}{5} MR^2 = \frac{2}{5} \div \frac{1}{4} \, I = \frac{8}{5} I$$

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Spherical shell:

$$\frac{2}{3} MR^2 = \frac{2}{3} \div \frac{1}{4} \, I = \frac{8}{3} I$$

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🔷 Step 3 — Total MOI of System

$$I_{total} = I + \frac{8}{5}I + \frac{8}{3}I$$

$$= \frac{15}{15}I + \frac{24}{15}I + \frac{40}{15}I$$
$$= \frac{79}{15} I$$

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🔷 Step 4 — Compare with Given Form

Given:

$$I_{total} = \frac{x}{15} I$$

So,

$$x = 79$$

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✅ Final Answer

$$\boxed{x = 79}$$

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⭐ Golden JEE Insight

Same mass & same radius → compare only coefficients.

Order of MOI:

$$\text{Shell} > \text{Solid sphere} > \text{Disc}$$

More mass away from center → more MOI.