Distance to Displacement Ratio in Circular Motion

❓ Question

A particle moves from $A$ to $B$ in a circular path of radius:

$$R$$

covering an angle:

$$\theta$$

as shown in figure.

Find the ratio of:

$$\text{Distance travelled : Magnitude of displacement}$$

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🖼 Question Image

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✍️ Short Explanation

This is a circular motion geometry problem.

👉 Distance travelled = arc length
👉 Displacement = straight line joining initial and final points
👉 Use circle geometry carefully.

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🔷 Step 1 — Distance Travelled 💯

Particle moves along circular arc.

Arc length formula:

$$s=R\theta$$

(where $\theta$ is in radians)

So distance travelled:

$$=R\theta$$

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🔷 Step 2 — Magnitude of Displacement

Displacement is chord $AB$.

Chord length formula:

$$AB=2R\sin\frac{\theta}{2}$$

So displacement magnitude:

$$=2R\sin\frac{\theta}{2}$$

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🔷 Step 3 — Required Ratio

$$\frac{\text{Distance}}{\text{Displacement}} = \frac{R\theta}{2R\sin\frac{\theta}{2}}$$

Cancel $R$:

$$= \frac{\theta}{2\sin\frac{\theta}{2}}$$

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🔷 Step 4 — Final Expression

Required ratio:

$$\boxed{ \frac{\theta}{2\sin\frac{\theta}{2}} }$$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Arc length ko displacement maan lena

❌ Degree measure directly formula mein use kar dena

❌ Chord formula bhool jaana

Remember:

$$\text{Distance}=\text{Arc length}$$
$$\text{Displacement}=\text{Chord length}$$

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✅ Final Answer

$$\boxed{ \frac{\theta}{2\sin\frac{\theta}{2}} }$$

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📚 Related Topics

• Charged Particle in Non-Uniform Magnetic Field
• Radius vs Energy in Magnetic Field
• Displacement of Charged Particle in Two B Fields