Acceleration means any change in velocity — and velocity is a vector, so it changes if either its size or its direction changes. An object moving in a circle at a perfectly constant speed is still accelerating, because its direction is changing every instant. This is the single idea that unlocks the whole topic.
At every point, the velocity vector is tangent to the circle (it points in the direction of motion). Because that direction is continuously turning, the velocity is continuously changing — so there must be an acceleration, and therefore a net force, causing it.
The acceleration in circular motion always points toward the center of the circle — perpendicular to the velocity. We call it centripetal acceleration ("center-seeking"). Its magnitude is:
where v is the object's speed and r is the radius of the circular path. Notice speed is squared: doubling the speed makes the centripetal acceleration four times larger, while doubling the radius only halves it.
A car travels at 20 m/s around a curve of radius 50 m. Find its centripetal acceleration.
By Newton's Second Law, an acceleration toward the center requires a net force toward the center. Combining ΣF = ma with a_c = v²/r gives the centripetal force equation:
This is not a new force. "Centripetal force" is just the name for the net inward force — whatever real force happens to point toward the center. It is the left side of Newton's Second Law applied to the radial direction.
Watch velocity (green) stay tangent and centripetal force (amber) always point inward. Change speed, radius, and mass — notice speed matters most because it enters the formula as v².
Try doubling speed — force quadruples. Then double radius — force only halves. Speed dominates because it's squared.
In every circular-motion problem, your first job is to ask: which real force is pointing toward the center? That force (or the net of several) is what equals mv²/r.
A 0.5 kg ball is whirled in a horizontal circle of radius 1.2 m at 6 m/s. Find the tension in the string (ignore gravity for the horizontal case).
In a vertical circle — a roller-coaster loop, a bucket of water swung overhead — gravity is always down, but "toward the center" changes direction as you go around. At the very top, both gravity and the normal force point downward, toward the center, so they add together:
The minimum speed at the top happens when the track can barely touch the object — the normal force drops to zero (F_N = 0). Then gravity alone provides the centripetal force:
A roller coaster loop has a radius of 8 m. What is the minimum speed a cart needs at the top to maintain contact with the track? Use g = 10 m/s².
The period T is the time for one complete lap. Since the object covers the circumference 2πr in one period at speed v:
Frequency f is the number of laps per second, and it is simply the reciprocal of the period: f = 1/T (measured in hertz).
A circular orbit is the ultimate example: a satellite is in constant free fall, and gravity is the centripetal force that keeps it curving around the Earth. Setting gravity equal to mv²/r is how you find orbital speeds — a direct bridge to later gravitation topics.
A satellite orbits Earth at a radius of 8.0 × 10⁶ m where g ≈ 6.2 m/s² at that altitude. Find its orbital speed and period.