0Definition
Uniform Circular Motion is the motion of a particle along a circular path with constant speed.
Key points:
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Speed is constant, but velocity is not (because direction changes).
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Acceleration exists because direction changes, called centripetal acceleration.
📌 Key Concepts
1️⃣ Centripetal Acceleration (aca_cac)
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Always towards the center of the circle.
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Formula:
ac=v2ra_c = \frac{v^2}{r}ac=rv2
Where:
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vvv = speed of the particle
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rrr = radius of the circle
2️⃣ Centripetal Force (FcF_cFc)
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Force required to keep the particle in circular motion.
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Always towards the center.
Fc=mac=mv2rF_c = m a_c = \frac{m v^2}{r}Fc=mac=rmv2
Where:
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mmm = mass of particle
Sources of centripetal force:
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Tension in string (like a ball on string)
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Gravity (like planets orbiting Sun)
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Friction (like a car on a curved road)
3️⃣ Angular Velocity (ω\omegaω)
ω=2πT=vr\omega = \frac{2 \pi}{T} = \frac{v}{r}ω=T2π=rv
Where:
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TTT = time period (time for one revolution)
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vvv = speed along circular path
4️⃣ Relation Between Linear and Angular Quantities
v=rωv = r \omegav=rω ac=rω2a_c = r \omega^2ac=rω2
📊 Key Formulas for UCM
| Quantity | Formula |
|---|---|
| Centripetal acceleration | ac=v2ra_c = \frac{v^2}{r}ac=rv2 |
| Centripetal force | Fc=mv2rF_c = \frac{mv^2}{r}Fc=rmv2 |
| Angular velocity | ω=vr=2πT\omega = \frac{v}{r} = \frac{2\pi}{T}ω=rv=T2π |
| Linear speed | v=2πrTv = \frac{2 \pi r}{T}v=T2πr |
🌍 Real-Life Examples
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Earth revolving around Sun
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Car turning in a circular track
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Stone tied to string whirled in a circle
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Satellite orbiting Earth
🧠 Key Points to Remember
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Speed is constant, velocity changes.
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Acceleration is centripetal, points to the center.
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Work done by centripetal force = 0 (force perpendicular to motion).
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Higher speed or smaller radius → more centripetal force needed.
