Uniform Circular Motion and Angular Speed

Uniform circular motion is motion in a circle at constant speed; its period T is the time for one revolution, its frequency f = 1 ÷ T, its linear speed v = 2πr ÷ T, and its angular speed ω = 2π ÷ T = 2πf (in radians per second), related by v = ωr. In brief: although the speed is constant, the velocity changes direction continuously, so the object accelerates toward the center. One revolution is 2π radians (360°). Angular speed is the same for all points on a rotating rigid body, while linear speed increases with the distance from the axis.

Wheels, planets, CDs and merry-go-rounds all move in circles. This guide defines the quantities used to describe circular motion, explains radians and the relation between linear and angular speed, compares the quantities in a table, works through numerical examples and corrects common mistakes.

What quantities describe circular motion?

• Radius r: the distance from the center of the circle to the object.
• Period T: the time for one complete revolution (seconds).
• Frequency f: the number of revolutions per second, f = 1 ÷ T, in hertz. Revolutions per minute (rpm) = 60 × f.
• Linear (tangential) speed v: the speed along the circular path: v = distance in one revolution ÷ T = 2πr ÷ T = 2πrf. The velocity vector is tangent to the circle and constantly changes direction.
• Angular displacement θ: the angle swept out, in radians.
• Angular speed ω: the rate of change of the angle: ω = θ ÷ t = 2π ÷ T = 2πf, in rad/s.

What is a radian?

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. The arc length is s = rθ when θ is in radians. A full circle is 2π rad = 360°, so 1 rad = 57.3°, and 180° = π rad. Converting: radians = degrees × π ÷ 180. Radians are the natural unit because they make formulas like s = rθ and v = ωr simple, without extra factors. The radian is a dimensionless unit (a ratio of two lengths), and in equations the “rad” can be dropped from the final unit (ω in s⁻¹).

How are linear and angular speed related?

Because s = rθ, dividing by time gives v = ωr. For a rotating rigid body, all points have the same ω, but points farther from the axis have a larger v: on a merry-go-round, a child at the edge moves faster than one near the center. In uniform circular motion, the speed v is constant, but the velocity changes direction, so there is an acceleration (the centripetal acceleration a = v² ÷ r = ω²r), directed toward the center. This is covered in the next article.

Circular motion quantities compared

Worked example: wheels and the Earth

1. Ball on a string. A ball moves in a circle of radius 0.50 m with a period of 2.0 s. f = 1 ÷ 2.0 = 0.50 Hz; ω = 2π ÷ 2.0 = 3.14 rad/s; v = ωr = 3.14 × 0.50 = 1.57 m/s (check: 2π × 0.50 ÷ 2.0 = 1.57 m/s).
2. Car tire. A car travels at 20 m/s on tires of radius 0.30 m (the speed of the car equals the speed of the rim relative to the axle if there is no slipping). ω = v ÷ r = 20 ÷ 0.30 = 66.7 rad/s. The frequency f = 66.7 ÷ 2π = 10.6 rev/s = 637 rpm.
3. Earth’s rotation. The sidereal day is 86,164 s, so ω = 2π ÷ 86,164 = 7.29 × 10⁻⁵ rad/s. At the equator (r = 6.378 × 10⁶ m): v = 7.29 × 10⁻⁵ × 6.378 × 10⁶ = 465 m/s. At latitude 60°, the radius of the circle is r cos 60° = half, so v = 232 m/s.
4. Arc length. A point on a wheel of radius 0.40 m turns through 90° (π/2 rad): arc s = rθ = 0.40 × 1.57 = 0.63 m.
5. Hour hand. It completes one revolution in 12 h = 43,200 s: ω = 2π ÷ 43,200 = 1.45 × 10⁻⁴ rad/s.

How was circular motion studied?

Ancient astronomers believed that circles were the perfect motion of the heavens, and Ptolemy’s model used circles upon circles. Christiaan Huygens derived the expression for centripetal acceleration in 1659 (published in 1673). The word “radian” was first used in print in 1873 by James Thomson, brother of Lord Kelvin, and the radian became an SI supplementary unit in 1960 (a derived unit since 1995).

Which mistakes are common?

Mistake 1: Saying that the velocity is constant in uniform circular motion

The speed is constant. The velocity changes direction, and so the object is accelerating.

Mistake 2: Using degrees in v = ωr

ω must be in radians per second.

Mistake 3: Confusing frequency and angular speed

ω = 2πf. A frequency of 1 Hz is an angular speed of 6.28 rad/s.

Mistake 4: Thinking that all points on a disc have the same linear speed

Their angular speed is the same, while their linear speeds are proportional to the distance from the axis.