Why circular motion needs a force
An object moving in a circle constantly changes direction, so it is accelerating even at a constant speed. That acceleration points toward the center of the circle and requires an inward force, the centripetal force:
- Centripetal acceleration: a = v² ÷ r
- Centripetal force: F = m × v² ÷ r
- Angular velocity: ω = v ÷ r, in radians per second
- Period: T = 2π r ÷ v
The force is provided by something real: tire friction for a car, tension for a ball on a string, gravity for a satellite.
Worked example
A 3,300 lb (1,497 kg) car takes a curve with a 150 ft (45.7 m) radius at 30 mph (13.4 m/s).
- Acceleration: 13.4² ÷ 45.7 = 3.93 m/s², or 0.40 g.
- Force: 1,497 × 3.93 = 5,889 N, which the tires must supply through friction.
- Angular velocity: 13.4 ÷ 45.7 = 0.293 rad/s.
- A full circle at that speed would take 21.4 seconds, or 2.8 rpm.
The effect of speed
Because force depends on the square of speed:
| Speed in the same curve | Lateral acceleration |
|---|---|
| 20 mph | 0.18 g |
| 30 mph | 0.40 g |
| 40 mph | 0.71 g |
| 50 mph | 1.11 g |
Ordinary tires on dry pavement grip up to about 0.8 to 0.9 g. At 50 mph, this curve would exceed the grip of most cars, and on a wet road the limit is much lower. Road engineers bank curves and post advisory speeds for this reason.
Centrifugal force
Passengers feel pushed outward in a turn. That centrifugal force is an apparent force that appears only in the rotating frame of the car. Seen from outside, the passengers simply tend to keep moving in a straight line, and the door or seat belt pushes them inward.
Other examples
- Satellites: gravity provides the centripetal force; in low Earth orbit, 7.8 km/s keeps the circle.
- Washing machines: the drum forces clothes into a circle, but water escapes through the holes.
- Roller coasters: loops are shaped so riders feel 3 to 5 g at most.