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.
Change the radius and angular speed and watch the velocity and centripetal-acceleration vectors.
Circular Motion is Accelerated!
A matching number in the speed box is not enough to treat a circle like straight uniform motion. Velocity is a vector, so a new tangent already means a new velocity, and without a center-pointing acceleration there is no place for the gravity that holds a satellite, the tension in a string, or the friction under a car in a curve. With radius r and angular speed ω the linear speed is v = rω, centripetal acceleration is a = v²/r = rω², and those sit on the same line as the period T = 2π/ω. The reason this chapter exists is not to recite the letters, but to lock in the habit of counting a change of direction as accelerated motion.
🎡 Why is there acceleration if speed is constant?
①Speed is constant but the direction of velocity keeps changing!
③This acceleration always points to the center → centripetal acceleration
④If centripetal force is cut, the body leaves on the tangent. The sketch shows tangent v only
⑤Examples of centripetal force: gravity on a satellite, tension in a string, friction on a curve
Visualizing Uniform Circular Motion
4 m
3 rad/s
seegongsik.com
Velocity and centripetal acceleration vectors of an object in uniform circular motion
A circular track sits in the middle of the figure, and a green point is marked at one fixed place on that circle. A gold arrow leaves the point along the tangent and writes v in m/s; a red arrow aims at the center O and writes a_c in m/s²; a dashed radius carries r in m. Sliding radius between 1 m and 8 m changes the size of the circle and the lengths of both arrows together. Sliding angular speed between 1 rad/s and 6 rad/s leaves the point where it is and only rewrites the v and a_c numbers. The point does not travel around the rim, so what you read is one frozen layout of vectors and the values attached to them.
💡 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
A constant speed still tempts people to declare that acceleration is absent. Acceleration is the time rate of change of velocity, and velocity includes direction, so it is already nonzero when the tangent is simply different from point to point on the circle. Draw the centripetal force along the travel direction and the net force cannot hold the path as a circle; once that force is gone, the tangent at that instant is the straight path that follows. The ω² in a = rω² is there because a steeper change of direction is counted as a larger size. Separate tangent from center first, then substitute into the formula.
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, f = 1T = ω2π
Period T is the time for one lap. Frequency f = 1/T
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
exam-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)