In uniform circular motion, acceleration occurs because the direction of velocity changes even when speed is constant. Accurately understand the difference between inertial and rotating frames.
Basic Knowledge
The acceleration of uniform circular motion (centripetal acceleration) always points toward the center of the circle.a = v²/r = rω² (v: speed, r: radius, ω: angular velocity)
Centripetal force: F = mv²/r = mrω²
Relationship between period and angular velocity: ω = 2π/T
Centripetal force is not a “new force”; it is the resultant of forces such as gravity, tension, the normal force, and friction acting in the centripetal direction.
Inertial force (centrifugal force): An apparent force felt by an observer in a rotating non-inertial frame (accelerating frame). Its magnitude is mrω², and its direction is opposite to the centripetal force (outward). Inertial force does not actually exist; from an inertial-frame viewpoint, only centripetal force acts.
Centripetal acceleration: a = v²/r = rω²
Centripetal force: F = mv²/r = mrω²
Angular velocity: ω = 2π/T = 2πf
Centrifugal force: magnitude mrω², direction outward (used only in a non-inertial frame)
Speed: v = 2πr/T = rω
Further Study (Background and Significance)
In circular-motion problems, it is important to state whether you are solving in an inertial frame or a non-inertial frame (rotating frame). In an inertial frame, use F = mv²/r with centripetal force = ma directed toward the center; in a non-inertial frame, add centrifugal force and set up a balance equation. Both methods give the same answer.
For circular motion in a vertical plane (such as a loop), the minimum speed at the highest point is frequently tested. From the condition N ≥ 0 at the highest point, mg ≤ mv²/r → v ≥ √(gr).
The maximum speed at which a car can turn: when friction provides the centripetal force, μmg ≥ mv²/r → v ≤ √(μgr). This is the principle behind the turning-speed limit of a car.
For a satellite in a circular orbit, gravity provides the centripetal force: GMm/r² = mv²/r → v = √(GM/r).
- Direction of centripetal acceleration: always toward the center
- Centripetal force = the central component of the net force (a combination of existing forces)
- Centrifugal force is a fictitious force used only in a non-inertial frame
- ω = 2π/T (T is the period)
- At the highest point of vertical circular motion: minimum speed with N ≥ 0
- The answer is the same in either an inertial or non-inertial frame
- There is acceleration even in “uniform” motion (the direction changes)
Points to Note (Common Confusions)
① Centripetal force is the resultant of real forces. There is no special force called “centripetal force.” ② Centrifugal force is only for non-inertial frames. Adding it to an inertial-frame equation causes double counting. ③ Even in uniform circular motion, velocity changes (its direction changes). Acceleration is not necessarily zero. ④ Distinguish angular velocity ω [rad/s] from rotation frequency f [Hz] (= 1/T).
Practice
- An object of mass 0.50 kg moves uniformly on a circle of radius 2.0 m with a period of 4.0 s. Find the centripetal force.
- Find the minimum speed at the highest point of vertical circular motion of radius r.
- Describe in one sentence the difference between solving circular motion in inertial and non-inertial (rotating) frames.