Circular and centripetal motion

centripetal force formulauniform circular motioncentrifugal forcebanked curve formulaa = v squared over r

Centripetal acceleration and force in both linear and angular form, plus the curve-speed results that follow when friction or a bank supplies them.

Speed in Circular Motion (v = 2πr/T)

v = \frac{2\pi r}{T}

Speed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per period.

Angular Velocity from Period

\omega = \frac{2\pi}{T}

One full revolution is 2π radians, so angular velocity is 2π divided by the period.

Centripetal Acceleration (a = v²/r)

a_c = \frac{v^2}{r}

Inward acceleration of an object moving in a circle at constant speed.

Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

Centripetal acceleration written in terms of angular velocity rather than linear speed.

Centripetal Force (F = mv²/r)

F_c = \frac{m v^2}{r}

Net inward force required to keep a mass moving in a circle at constant speed.

Maximum Speed on a Flat Curve

v_{\max} = \sqrt{\mu_s g r}

Fastest a vehicle can round a flat, unbanked curve before friction can no longer supply the centripetal force.

Banked Curve Angle

\theta = \arctan\!\left(\frac{v^{2}}{r g}\right)

Bank angle that lets a vehicle round a curve of a given radius at a given speed with no reliance on sideways friction.

How they fit together

Nothing here is a new law. An object moving in a circle at constant speed is still accelerating, because its direction keeps changing, and that acceleration always points at the centre with magnitude v²/r. Feed it into F = ma and you get the centripetal force. The angular forms, a = ω²r and v = 2πr/T, are the same statement written for something that reports its motion in revolutions rather than metres per second.

Pick the linear form when you know a tangential speed, the angular form when you know a period or an rpm — converting rpm to ω first is usually less error-prone than converting to v. The deep mistake is treating centripetal force as an extra force to be added to the free-body diagram. It is not: it is the name for whatever real force already points inward — tension, friction, gravity, the normal force from a banked road. That is why the flat-curve and banked-curve results contain no mass at all; the mass cancels, so a loaded truck and an empty one slide at the same speed. And centrifugal force does not appear in any of these, because in an inertial frame it does not exist.