Tie a ball to a string and swing it in a circle. The string provides the force that keeps the ball from flying off in a straight line. That force, directed toward the center of the circle, is called centripetal force. It is not a new type of force. It is simply the net force directed inward that causes circular motion. For the ball on a string, tension provides the centripetal force. For a car turning a corner, friction provides it. For the Moon orbiting Earth, gravity does.
The formula is F equals mv squared over r, where m is mass, v is tangential speed, and r is the radius of the circle. Notice that the force increases with the square of the speed. Double the speed, and you need four times the force. This is why sharp turns at high speed are dangerous. The tires may not have enough friction to provide the needed centripetal force, and the car skids outward.
Our Centripetal Force Calculator lets you experiment with different masses, speeds, and radii. Try plugging in the values for a 1,000 kg car going 20 m/s around a curve with a 50-meter radius. The result is 8,000 newtons of centripetal force. If the tires can only provide 6,000 newtons of friction, the car slides.
Centripetal acceleration equals v squared over r. An object moving at 30 m/s in a circle of radius 100 meters experiences 9 m/s² of acceleration, nearly as much as gravity. Fighter pilots experience much more. In a 5g turn, the centripetal acceleration is five times gravity, enough to cause a pilot to black out if not wearing a g-suit that squeezes blood back toward the brain.
A common confusion is between centripetal and centrifugal force. Centripetal force is real. Centrifugal force is a fictitious force that appears in a rotating reference frame. If you are in a spinning carnival ride, you feel pushed outward. From the ground, it is clear that your body is simply trying to move in a straight line while the ride forces you into a circle. The outward feeling is inertia, not a real force.
Satellites stay in orbit because gravity provides the centripetal force needed for circular motion. The ISS at 408 km altitude moves at about 7.66 km/s. Gravity at that altitude is still about 8.7 m/s², and this is exactly the centripetal acceleration needed for that speed at that radius. This is not a coincidence. It is the definition of a stable circular orbit.