An object moving along a circular path constantly changes direction, even if its speed never changes — and any change in velocity means acceleration. Centripetal acceleration is the component of that acceleration directed toward the center of the circular path.
Centripetal acceleration doesn't need a new, special "centripetal force" — it's produced by whatever combination of already-familiar forces happens to point toward the center. Three classic scenarios show this in action.
At the very top of a vertical circular loop, there's a minimum speed required to keep an object in contact with the track. At exactly this speed, normal force drops to zero and gravity alone provides all the centripetal force needed:
On a banked curve, both the normal force and static friction can contribute components toward the center. At one particular "ideal" speed, the banking angle alone supplies exactly the centripetal force needed — no friction required at all.
A mass swinging in a horizontal circle at the end of a string (tracing out a cone) relies on a component of tension to provide its centripetal force, while the rest of the tension balances gravity vertically.
Explore the vertical loop and banked curve scenarios directly below.
Two classic centripetal-force scenarios. Toggle between them and explore what each requires.
At exactly this speed, gravity alone provides all of the centripetal force needed — normal force from the track drops to zero right at the top.
A 2 kg mass swings in a horizontal circle of radius 0.5 m at the end of a 1.2 m string, tracing a cone with the string making a constant angle θ with the vertical. Find the tension in the string and the object's speed.
Everything above assumed constant speed around the circle. If speed is also changing, a second acceleration component appears: tangential acceleration — the rate at which speed itself changes, directed tangent to the circular path (along the direction of motion, not toward the center).
An object's true, total acceleration in circular motion is the vector sum of both pieces:
Explore both components together below.
An object moving in a circle at 12 m/s, radius 8 m. Set the tangential acceleration (positive = speeding up, negative = slowing down) — centripetal acceleration never goes away, but tangential acceleration only appears when speed is changing.
For an object in uniform circular motion (constant speed), the motion repeats itself every revolution — a natural fit for describing it with period (T, the time for one full revolution) and frequency (f, how many revolutions happen per unit time).
Since the object travels one full circumference, 2πr, during each period, at constant speed v:
A satellite in a circular orbit has its centripetal acceleration provided entirely by gravitational attraction from the central body — combining Newton's law of universal gravitation (Lesson 2.6) with the centripetal acceleration equation from this lesson connects a satellite's orbital period directly to the mass of whatever it's orbiting:
Derive T² = 4π²R³/(GM) starting from Newton's law of universal gravitation and the centripetal acceleration equation, for a satellite of mass m orbiting a central body of mass M at radius R.