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An object moving in a circle at constant speed is nonetheless accelerating because its direction changes. The centripetal acceleration points toward the center of the circle and has magnitude a_c = v²/r = ω²r, where ω is the angular velocity. Period T (time per revolution), frequency f = 1/T, and angular velocity ω = 2π/T are the key kinematic parameters.
Derive the centripetal acceleration formula geometrically by computing the change in velocity vector over a small arc. Practice converting between linear quantities (v, a) and angular quantities (ω, α) using the arc-length relationships v = ωr and a_t = αr.
From 1D kinematics you learned that acceleration means changing speed. Circular motion challenges this intuition: an object can accelerate while its speed stays perfectly constant. The key is remembering that velocity is a vector — it has both magnitude (speed) and direction. When an object moves in a circle, its speed may be fixed, but its direction of motion changes continuously. Any change in velocity, whether in magnitude or direction, constitutes acceleration.
To see this concretely, imagine a car moving clockwise around a circular track. At the top of the circle the car moves to the right; a quarter-turn later it moves downward. The velocity vector has rotated 90°. The change in velocity over that quarter-turn points inward — toward the center of the circle — and dividing by the time elapsed gives the centripetal acceleration. Working through the geometry carefully (comparing velocity vectors at two nearby points and taking the limit as the arc shrinks) yields the formula a_c = v²/r = ω²r, always directed toward the center.
The three key kinematic parameters — period T, frequency f, and angular velocity ω — are tightly linked. T is the time for one complete revolution; f = 1/T is revolutions per second (Hz); ω = 2π/T is radians per second. The arc-length relation s = rθ connects angular and linear quantities: differentiating gives v = ωr (linear speed equals angular speed times radius), and differentiating again gives the tangential acceleration a_t = αr when angular speed is changing. For uniform circular motion (constant speed), α = 0 and only the centripetal acceleration exists.
It helps to keep centripetal and tangential acceleration clearly distinct. Centripetal acceleration (a_c = v²/r) points radially inward and is responsible for changing the *direction* of velocity. Tangential acceleration (a_t = αr) points along the arc and is responsible for changing the *magnitude* of velocity (speeding up or slowing down). In uniform circular motion there is only centripetal acceleration. In non-uniform circular motion (a car speeding around a curve) both components are present simultaneously, and the total acceleration vector is their vector sum.
These kinematic quantities describe the *motion* without specifying the *cause*. What produces the centripetal acceleration — a tension, friction, gravity, or a normal force — is the subject of circular motion dynamics. But understanding the kinematics first is essential: before you can apply Newton's second law to circular motion (F_net = ma_c = mv²/r), you must be clear that the required acceleration exists, is nonzero even at constant speed, and always points toward the center.