🌍Uniform Circular Motion
Moving in a Circle
Uniform circular motion is movement along a circular path at constant speed. Note: speed is constant, but velocity is NOT — because direction is always changing.
Since velocity is changing, there must be acceleration! This acceleration always points toward the center of the circle and is called centripetal acceleration ("centripetal" means "center-seeking").
Key quantities: - Period (T): time to complete one full revolution (seconds) - Frequency (f): revolutions per second (Hz) — f = 1/T - Circumference: 2πr
Speed: v = 2πr/T (distance per revolution divided by time per revolution)
Centripetal Acceleration
The centripetal acceleration always points toward the center:
a_c = v²/r
where v is speed and r is radius of the circle.
This acceleration is always perpendicular to velocity — it changes the direction of velocity, not its magnitude. That's why the speed stays constant even though the object is accelerating!
Think of it this way: if you swing a ball on a string in a circle, the string always pulls the ball toward your hand (the center). Let go of the string and the ball flies off in a straight line — tangent to the circle.
Circular Motion Visualization
Watch the velocity vector (blue, tangent) and centripetal acceleration (red, inward) as the object moves in a circle.
Think About It
A car goes around a curve. Is there a centripetal acceleration? If so, what real force provides it?
✏️ Worked Example
Problem: A 1000 kg car travels around a circular track of radius 50 m at 20 m/s. What is the centripetal acceleration?
📐 Key Equations
Uniform circular motion
v = (2π r)/(T)a_c = (v²)/(r) = (4π² r)/(T²)f = (1)/(T)⚠️ Common Mistakes
Misconception: Confusing speed and velocity in circular motion — "speed is constant so nothing is changing."
✓ Correct thinking: Speed is constant but velocity is not. Velocity is a vector, and its direction changes continuously around the circle.
Why: Changing direction means changing velocity, which means there is acceleration — even at constant speed. This is the centripetal acceleration.
Misconception: Thinking centripetal acceleration points outward (away from the center).
✓ Correct thinking: Centripetal acceleration always points inward, toward the center of the circle.
Why: The net force (and therefore acceleration) must point toward the center to continuously redirect the velocity vector inward. "Centripetal" literally means center-seeking.
Misconception: Using T (period) where f (frequency) is needed, or vice versa.
✓ Correct thinking: Period T is in seconds per revolution; frequency f is in revolutions per second (Hz). They are reciprocals: f = 1/T.
Why: Mixing up T and f in the formula v = 2πr/T gives an answer that is off by a factor of T², often by many orders of magnitude.
📝 Practice Problems
Try these problems. Check your answer when ready.
A ball moves in a circle of radius 3 m with a period of 2 s. What is its speed?
v = (2π r)/(T)A toy car travels around a circular track of radius 0.5 m at 2 m/s. What is its centripetal acceleration?
a_c = (v²)/(r)A satellite orbits at constant speed. Is it accelerating? Explain.
A runner completes a circular track of radius 40 m in 25 s. Find (a) the runner's speed and (b) centripetal acceleration.
v = (2π r)/(T), a_c = (v²)/(r)A point on the edge of a spinning disk (r = 0.2 m) has centripetal acceleration of 50 m/s². What is the frequency of rotation?
a_c = 4π² r f²Two objects move in circles at the same speed. Object A has radius 2 m and object B has radius 8 m. Compare their centripetal accelerations.
A car rounds a curve of radius 60 m. What maximum speed can it travel if the centripetal acceleration must not exceed 0.5g (≈ 5 m/s²)?
a_c = (v²)/(r)An object moves in a horizontal circle of radius r. Its period is doubled while the radius stays the same. By what factor does the centripetal acceleration change?
a_c = (4π² r)/(T²)Finished reading through this lesson?