12.2 Centripetal acceleration
- Syllabus
- 9702–2028–2029
- Topic
- 12.2
- Level
- A2
Learning objectives
12.2.1A force of constant magnitude that is always perpendicular to the direction• understand that a force of constant magnitude that is always perpendicular to the direction of motion causes centripetal acceleration12.2.2Centripetal acceleration causes circular motion with a constant angular• understand that centripetal acceleration causes circular motion with a constant angular speed12.2.3A = rω2 and a = v2 / r• recall and use a = rω2 and a = v2 / r12.2.4F = mrω2 and F = mv2 / r• recall and use F = mrω2 and F = mv2 / r
A continuously perpendicular resultant turns velocity toward a circle
A resultant force of constant magnitude that is always perpendicular to the instantaneous velocity produces an acceleration in the force direction. For circular motion this direction is radially inward, toward the centre.
The force has no component along the velocity, so it does no work and does not change kinetic energy or speed. During each short interval it changes the velocity vector toward the centre; as the force direction turns continuously, the path curves into a circle.
| Situation | Real force or component supplying inward resultant |
|---|---|
| stone on a string | tension |
| satellite orbit | gravity |
| car on a level bend | friction |
| ball on a banked surface | inward component of normal contact force |
Zero work does not mean zero force or acceleration: speed can remain constant while velocity direction changes. One fixed-direction perpendicular impulse is insufficient—the resultant must remain perpendicular as the object moves.
In uniform circular motion, velocity is tangent and acceleration is inward
| Quantity | Magnitude | Direction during motion |
|---|---|---|
| velocity v | constant | tangent to the circle; continually changes |
| centripetal acceleration a | constant for fixed r and speed | toward the centre; continually changes |
| angular speed ω | constant | rotation proceeds through equal angles in equal times |
At every point, centripetal acceleration is perpendicular to instantaneous velocity. It continuously turns the velocity vector, producing a circular path while its constant magnitude maintains uniform angular speed.
After one complete revolution, the object's position and velocity direction return to their starting values, but throughout the revolution velocity, acceleration and resultant-force directions have varied.
Constant speed is only constant magnitude of velocity, not constant velocity. Nonzero inward acceleration is required even though there is no tangential acceleration in uniform circular motion.
Centripetal acceleration has magnitude a=rω²=v²/r toward the centre
For circular motion, centripetal acceleration is a=rω²=v²/r and points radially inward.
Use angular speed or tangential speed consistently, and remember the acceleration is perpendicular to instantaneous velocity in uniform motion.
At r=0.50 m and ω=4.0 rad s⁻¹, a=rω²=8.0 m s⁻².
The object is accelerating even at constant speed because its velocity direction changes.
The centripetal resultant force is F=mrω²=mv²/r
The inward resultant force needed for circular motion is F=ma=mrω²=mv²/r.
Identify which real interaction supplies it—tension, gravity, friction or a normal force—and sum forces toward the centre.
A 2.0 kg mass moving at 3.0 m s⁻¹ around radius 1.5 m needs an inward resultant of 12 N.
Centripetal force is not an additional force to draw; it is the name for the inward resultant.