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Uniform Circular Motion

Uniform Circular Motion

In uniform circular motion the speed is constant but the direction of velocity keeps changing, so there is acceleration. That acceleration always points to the center: centripetal acceleration a = v²/r = rω². If the centripetal force vanishes, the object flies off along the tangent. Slide radius and angular speed here to watch the velocity and centripetal-acceleration vectors change.

Circular Motion is Accelerated!
🎡 Why is there acceleration if speed is constant?
①Speed is constant but the direction of velocity keeps changing!
②Direction change → velocity changes → acceleration exists
③This acceleration always points to the center → centripetal acceleration
Visualizing Uniform Circular Motion
4 m
3 rad/s
💡 Key Observations
①Velocity (v) is always tangent to the circle
②Centripetal acceleration (a) always points to the center
③Larger r or ω → larger v and larger centripetal acceleration
Uniform Circular Motion Formulas
Centripetal Acceleration
a = v²r = rω²
Acceleration toward center: v²/r = rω²
Linear Velocity
v = rω
v (m/s) = r (m) × ω (rad/s)
Period and Frequency
T = 2πω = 2πrv
Time for one revolution
Centripetal Force
F = ma = mv²r = mrω²
Force needed to maintain circular motion (toward center)
Worked Examples
Example 1
An object in uniform circular motion on a 2 m radius moves at 4 m/s. What is its centripetal acceleration?
1
Use the centripetal acceleration formula a = v²/r.
a = v2r
2
Substitute v = 4 m/s, r = 2 m.
a = 422 = 8 m/s2
▸ 8 m/s² (toward the center)
Centripetal acceleration always points to the center. Even at constant speed, the changing direction means there is acceleration.
Example 2
A 0.5 kg object on a 1 m string moves in a horizontal circle at angular speed 2 rad/s. What is the string tension?
1
The tension provides the centripetal force. F = mrω².
F = mrω2
2
Substitute m = 0.5 kg, r = 1 m, ω = 2 rad/s.
F = 0.5 × 1 × 22 = 2 N
▸ 2 N
For a string circle, tension = centripetal force. Use F = mrω² or mv²/r depending on the given quantities.
Summary
Centripetal acceleration
a = v²r
Linear velocity
v = rω
CSAT-style
In uniform circular motion, if the radius is kept fixed and the angular speed is doubled, by what factor does the centripetal acceleration change?
①2×
②4×
③12×
④No change
⑤8×
▸ ② 4×
1
Centripetal acceleration is a = rω², proportional to the square of the angular speed.
a = rω2
2
If ω doubles, ω² quadruples → a is 4×.
ω→2ω ⇒ a ∝ (2ω)2 = 4rω2
🎯 Exam Points
①Uniform circular: constant speed, changing direction → acceleration exists
②a = v²/r = rω² (always toward center)
③F = mv²/r — without force, the object flies off in a straight line
④v = rω, T = 2π/ω, f = 1/T = ω/(2π)
⑤Centripetal force examples: gravity (satellites), tension (ball on string), friction (cars in curves)
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Gravitation and Kepler's Laws
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