Centrifugal and Centripetal Force
Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay
Centrifugal Force
Here's a common situation. You're riding in a car when the driver turns sharply to the left. You get crunched into the door, and objects on the dashboard go skittering across the dash. We commonly call this centrifugal force.
But where does the force come from? The driver didn't lay a hand on you. Nobody touched the object on the dash. What made it move? Why did it start acting the instant the car started to turn?
As an everyday description of what happens when things turn, centrifugal force works reasonably well. But when we try to describe phenomena in detail, or do detailed calculations, it breaks down. We have a mysterious force that comes out of nowhere and disappears just as mysteriously. There's no connection between the object on the dash and the car, so what pushes it? Why does this mysterious force act only on objects in the car?
Centripetal Acceleration
Now let's look at it from the perspective of somebody outside the car. The car turns left. Objects not firmly attached to the car, like the object on the dash, aren't "pushed" at all. Instead, the object keeps on moving in a straight line while the car turns out from under it.
Your head is also not firmly attached to the car and tries to do the same thing. However, your derriere is attached to the car, by friction with the seat and (we hope) by a seat belt. What your body feels is that you tilt to the right. What really happens is that your rear end moves leftward out from under your head. As the turn progresses, eventually the car door turns inward enough to collide with your head. You, of course, feel this as your head pressing into the door.
The only real forces here, the only ones physically pushing on any objects, are all directed into the turn, not outward. We have friction between the tires and the road pushing the car to the left because the front wheels are turned. We have friction with the seat and the seat belt forcing your posterior to go along with the car as it turns. Finally we have the car door pushing your head in the direction of the turn.
This is why scientists don't speak of "centrifugal force" very much, except loosely. When they really have to be exact in their descriptions, they speak of "centripetal force." The "petal" part comes from a Latin word meaning to seek (we find the same root in the word "petition") so centripetal means "center-seeking."
But if you close your eyes as the car turns, it doesn't feel like you're moving in a straight line. You feel the turn. Why? Because your inner ear has a little inertial guidance system: a small chamber called the cochlea filled with fluids and some tiny loose grains. As these grains move, they bump against sensor cells that transmit signals to the brain, where they are interpreted as a feeling of motion. (This system helps us maintain our balance. Upsets to it, like an infection, can cause it not to work, which is why you can sometimes find it hard to maintain your balance when you're sick. Also, violent motion can send so many conflicting signals that your brain can't make sense of them, and takes a while to sort things out. Think Tilt-a-Whirl.) Your body also has a kinesthetic sense. You can tell where your body parts are even with your eyes closed.
So, when you go into a turn with your eyes closed, your inner ear behaves like the object on the dash in miniature. Your head and inner ear turns but the little grains keep going straight until they bump into the sensor cells. Also, your head and seat try to travel different paths and your kinesthetic sense reports this to the brain.
Calculating the Force
Actually what we want is acceleration, which is change in velocity. Force equals mass times acceleration. The force the your inner ear exerts when it stops that tiny grain from moving is a lot smaller than the force the car door exerts on your head, but the acceleration is the same. Acceleration can result from a change in speed, or direction, or both.
| Here's your car making the turn. It's traveling in a circular arc of radius r at constant speed v. But the direction it's traveling changes. The velocity is a vector - it has both size (the speed) and direction (always at right angles to the radius). But the direction of the velocity changes, so there's an acceleration. We show the velocity and radius at two instants in different colors. |
 |
| The acceleration a is the change in velocity and is shown by the green arrow. How big is it? It's easy to see that the angle between the two radii and the angle between the two velocity vectors has to be the same. Call it O. So we have two similar triangles, with a/v = v/r, and a = v2/r. |
 |
If you measure speed in terms of time of revolution, then v = 2(pi)r/t. The velocity is the circumference of the circle divided by the time. Since a = v2/r, we can substitute 2(pi)r/t for v and get a = 4(pi)2r/ t2 = 39.478r/ t2.
In the Middle Ages, one argument against the rotation of the earth was that people would fly off. They didn't know how to calculate the acceleration (in fact they didn't know what to calculate - it took a long time to sort out the concepts of velocity, momentum, force and acceleration). If you visit the Q/a site Quora, you will find people who, incredibly, still ask why we can't feel the earth rotating.
But we can. The radius of the earth is 6,400,000 meters, and it makes one revolution per day (86,400 seconds). So the acceleration at the equator is 4(pi)2*6,400,000/ 86,4002 = 0.033 meters per second per second. Since the normal acceleration of gravity is 9.8 meters per second per second, the effect is tiny. But measurable. It affects, for example, how well pendulum clocks keep time. Basically at the equator 0.033 meters per second per second of gravity goes into just keeping things on the earth, overcoming the tendency to fly off at a tangent, leaving that much less to affect weights on scales.
What value would t have to have for the acceleration to equal that of gravity? Set a = 9.8 and solve for t. You get 5077 seconds or 1.4 hours. At that point, gravity would just be sufficient to provide enough centripetal acceleration to keep things tethered to the earth.
Wait a Minute...
Roller Coasters
Okay, if I'm traveling in a circle and the acceleration is inward, what happens in a roller coaster with a loop? At the top of the loop I'm accelerating downward, so why don't I fall out?
Remember: you're not accelerating. The seat is, and it's pushing you with it. If your harness fails, you'll go flying off tangent to the arc even though your seat continues to travel in an arc. To get to the top of the loop, you went up at a pretty high rate of speed, so if you went loose at this point, you'd still continue upward. But the seat prevents that from happening. At the top of the loop, your inertia tends to keep you moving in a straight line (horizontally by now), and gravity wants to pull you down (at 9.8 meters per second per second). If the acceleration of the roller coaster is equal to gravity, the seat will accelerate downward just as fast as you are. You still stay in contact with the seat. And this is only for a split second because your direction is constantly changing. Since you also have a pretty good forward velocity, you'd actually travel a parabolic path if you flew loose from the coaster. But the coaster travels an arc with sharper curvature and scoops you up.
In reality most coasters are designed to exceed the acceleration of gravity in loops, so the seat is moving inward faster than gravity can pull you. However, for thrills, it might just pay at times to let the acceleration be a hair less than gravity to give you a feeling of falling. You are of course, held by a harness and the people who design these things are very, very, good at understanding centripetal acceleration.
At the top of an upright roller coaster, your velocity plus gravity would cause you to travel a parabolic path if you flew free. If the actual arc of the coaster equals the arc of the parabola, the seat drops just as fast as you do and you are effectively in free fall for a second or two. Water slides where it feels like you leave contact with the slide for a split second employ the same principle.
Incidentally, your innards feel funny in these situations because it takes them a split second to catch up with your body.
Driving on Curves
Race drivers accelerate going around turns. Wouldn't that cause them to lose control?
Remember, a = v2/r. So if you increase velocity, you increase centripetal acceleration. If you have a good grip on the road, accelerating helps you make the turn. Also, since most curves are banked, your inertial tendency to fly off at a tangent will push you into the pavement, and increasing your velocity aids the process.
On the other hand, if there's anything to disconnect you from the road (oil, ice, water, loose gravel) then the only thing you've got is your inertia and you will go flying off tangent to the arc. Also, there's a limit to the amount of frictional force between tires and pavement. Create centripetal acceleration that requires more force than you've got holding you to the road, and inertia takes over. So accelerating in a turn works only up to a point.
Return to Earth Science Notes Index
Return to Physical Geology Notes Index
Return to Professor Dutch's home page
Created 8 February, 1997
Last Update 23 January, 2001
Not an official UW-Green Bay site