Centripetal Force (Circular Motion)

Insert mass, orbital speed and radius into F_z = m·v²/r. Example: a body of 0.5 kg circles at 4 m/s on a radius of 2 m: F_z = 0.5 × 16 / 2 = 4 N. Note that v is squared; double the speed demands four times the force. If the period T or the rotation rate is given instead of the speed, use the form F_z = m·ω²·r with ω = 2π/T. The force always points to the centre of the circular path. Check the units: mass in kg, speed in m/s, radius in m, and the force comes out in newtons.

The centripetal force is the real, inward-directed force that keeps a body on the circular path, supplied for example by string tension, friction or gravity. The centrifugal force, by contrast, is a fictitious force: it exists only for observers rotating along and describes their sensation of being pushed outward. Physically, something else happens in the resting frame: the body "wants" to continue straight ahead (inertia), and the centripetal force constantly bends it onto the curve. If the string snaps, the body does not fly radially outward but continues straight along the tangent, the classic test of whether the concept is understood. In free-body diagrams only the centripetal force (or its real sources) appears.

The centripetal force is not a separate force of nature but a role description; some real force must play it. For a hammer thrower it is the string tension, in cornering the static friction between tyres and road, for a satellite gravity, for an electron in a magnetic field the Lorentz force, in a banked curve the normal force component. Exactly this assignment is the core of many problems: you set the available real force equal to m·v²/r. Cornering example: static friction µ·m·g = m·v²/r yields the maximum cornering speed v = √(µ·g·r), independent of mass. If the real force is insufficient, the body leaves the circular path outward (the car slides straight on).

Via the angular velocity ω. It relates to the period T through ω = 2π/T and to the rotation rate n through ω = 2π·n; the orbital speed is v = ω·r. The centripetal force then becomes F_z = m·ω²·r. Carousel example: a person (50 kg) sits at r = 5 m while the carousel turns at ω = 1.2 rad/s: F_z = 50 × 1.44 × 5 = 360 N. This form reveals an important subtlety: at fixed angular velocity the force grows linearly with radius, so you sit "harder" on the outside. At fixed orbital speed it is the other way round: F_z = mv²/r decreases with larger radius. The given quantities decide which form to choose.

Because acceleration means any change of the velocity vector, and that includes direction. On a circular path the direction of motion turns continuously even if the speedometer stays constant. This change of direction is the centripetal acceleration a_z = v²/r, always pointing to the centre. Without it there would be no curve: by Newton first law a force-free body travels straight. You feel this in a car: in a tight curve at 50 km/h (13.9 m/s) and r = 30 m, a_z = 13.9²/30 ≈ 6.4 m/s², about two thirds of g, and you are pulled noticeably sideways. Mnemonic: constant speed does not mean constant velocity.