1aCircular Motion
Analyze centripetal acceleration and force, angular velocity, period, and frequency for objects moving in circular paths.
Circular motion is the foundation for understanding planetary orbits, centrifuges, and any rotating system. Mastering centripetal force is essential before studying gravity and orbital mechanics.
Why does a car need a larger centripetal force to round a tight curve at high speed — and what happens when that force isn't enough?
Lesson Overview
When an object moves in a circle at constant speed, its velocity direction changes continuously — meaning it accelerates even though its speed stays the same. This centripetal acceleration always points toward the center of the circle and requires a net inward force called the centripetal force. We describe circular motion using period (T), frequency (f), and angular velocity (ω), and connect them to tangential speed through v = ωr.
Key Equations
Worked Examples
A car rounds a curve of radius r = 50 m at v = 20 m/s. (a) Find the centripetal acceleration. (b) Find the centripetal force if m = 1200 kg.
An object completes one full revolution in T = 4 s. (a) Find ω. (b) Find tangential speed if r = 3 m.
A satellite orbits Earth at r = 7 × 10⁶ m with v = 7546 m/s. Find the orbital period T.
A 0.5 kg ball on a 1.2 m string moves in a horizontal circle at v = 6 m/s. Find the tension in the string (which provides the centripetal force).
A wheel completes 300 revolutions per minute (RPM). Find (a) its frequency in Hz, (b) its period T, and (c) its angular velocity ω.
Guided Problems
A ball on a string moves in a horizontal circle of radius r = 0.8 m at v = 4 m/s. Find the centripetal acceleration.
Hint: Use aₒ = v²/r. Square the speed first, then divide by the radius.
A merry-go-round completes one revolution every T = 6 s. Find its angular velocity ω and the tangential speed of a child sitting r = 2 m from the center.
Hint: First find ω = 2π/T, then use v = ωr.
A 1500 kg car rounds a flat curve of radius r = 80 m at v = 25 m/s. Find the centripetal force required.
Hint: Use Fₒ = mv²/r. This force is provided by friction between the tires and the road.
A point on a rotating disk is r = 0.3 m from the center. The disk has ω = 10 rad/s. Find the tangential speed of that point.
Hint: Use v = ωr. Multiply the angular velocity by the radius.
An object moves in a circle of radius r = 5 m with a centripetal acceleration of aₒ = 20 m/s². Find its speed v.
Hint: Rearrange aₒ = v²/r to get v = √(aₒ × r). Then take the square root.
Key Vocabulary
Uniform Circular Motion
Motion in a circular path at constant speed. Although speed is constant, velocity changes direction continuously, so the object is always accelerating.
Example: A satellite in a circular orbit moves at constant speed but constantly changes direction.
Centripetal Acceleration
The acceleration directed toward the center of a circular path. Its magnitude is aₒ = v²/r. It changes the direction of velocity, not its magnitude.
Example: A car at 20 m/s on a 50 m radius curve has aₒ = 8 m/s² pointing toward the center.
Centripetal Force
The net inward force required to keep an object moving in a circle: Fₒ = mv²/r. It is not a new type of force — it is the name for whatever real force (friction, tension, gravity) acts centripetally.
Example: Friction from the road provides the centripetal force that keeps a car on a curved road.
Period (T)
The time for one complete revolution, measured in seconds. Related to frequency by T = 1/f.
Example: A satellite with T = 5828 s takes about 97 minutes to complete one orbit.
Frequency (f)
The number of complete revolutions per second, measured in hertz (Hz). f = 1/T.
Example: A wheel completing one revolution every 0.2 s has f = 5 Hz.
Angular Velocity (ω)
The rate of change of angular position, measured in radians per second (rad/s). ω = 2π/T = 2πf. Relates to tangential speed by v = ωr.
Example: An object with T = 4 s has ω = π/2 ≈ 1.57 rad/s.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
A car travels at v = 15 m/s around a curve of radius r = 45 m. What is the centripetal acceleration?
An object completes one revolution in T = 2 s. What is its angular velocity?
A 2 kg object moves in a circle of radius 4 m at v = 6 m/s. What centripetal force is required?
A wheel has ω = 6 rad/s. What is the tangential speed at r = 0.5 m from the center?
Which direction does centripetal acceleration point?
Independent Practice
A car travels around a circular track of radius 50 m at 20 m/s. Calculate (a) the centripetal acceleration and (b) the centripetal force if m = 1200 kg.
A wheel rotates at 300 RPM. Convert to rad/s and find the linear speed of a point 0.40 m from the center.
A 2.0 kg ball on a 1.5 m string is swung in a horizontal circle at 4.0 m/s. Find the tension in the string.
A satellite orbits at r = 6.8 × 10⁶ m with T = 5580 s. Find its orbital speed v = 2πr/T.
★ A 1000 kg car rounds a flat curve of radius r = 60 m. The coefficient of static friction μₛ = 0.7 and g = 10 m/s². Find the maximum safe speed. Show all steps.
ChallengeCommon Mistakes
Calling centripetal force a separate 'extra' force on the free-body diagram
Centripetal force is the NET inward force provided by real forces (tension, gravity, normal force) — never draw it as a separate arrow on an FBD
Confusing angular velocity ω (rad/s) with linear speed v (m/s)
They are related by v = ωr. Angular velocity describes how fast the angle changes; linear speed depends on the radius
Forgetting to convert RPM to rad/s before using rotational equations
Convert: ω (rad/s) = RPM × 2π/60. Always use radians in rotational formulas
Thinking an object moving at constant speed in a circle has zero acceleration
Speed is constant but velocity direction changes continuously — the object has centripetal acceleration directed toward the center
Math Tips
Centripetal acceleration: aₒ = v²/r. The net inward force equals maₒ = mv²/r. Identify which real force(s) provide this inward push.
Period and frequency are reciprocals: T = 1/f. Angular velocity ω = 2π/T = 2πf. Always convert RPM to rad/s using ω = RPM × 2π/60.
Tangential speed v = ωr. A point farther from the center moves faster even though ω is the same for all points on a rigid body.
For circular orbit problems, set the centripetal force equal to the available force (gravity, friction, tension) and solve for the unknown.