Centrifugal force and centripetal force

A spinning carousel

The barfing bowl

The local amusement park of my childhood had a ride called the Mixing Bowl. Riders would sit around the periphery of a giant bowl. The bowl would then spin, and riders could feel the centrifugal force pushing them out against the side of the bowl. Some would struggle to crawl to the center. As one got closer to the center, it would get easier; the centrifugal force would weaken. Those who made it to the center could sit and relax. Less robust riders would vomit.

A physics teacher told me that centrifugal force is a fictitious force. The real force, called centripetal force, is actually pulling the riders in toward the center. This made no sense to me. The struggle needed to crawl to the center is quite real, so the rider must be fighting an outward force. What's going on here? We can better understand the teacher's claim by starting with the straight line motion of an airplane taking off.

Airplane engines powering takeoff acceleration

Airplane takeoff acceleration

Imagine the start of an airplane's takeoff acceleration. What would happen if we had placed a tennis ball in the aisle? We would see it accelerate toward the rear of the aircraft. Newton's Second Law says that when there is acceleration, there must be a force causing it. So, we would conclude that there must be a force pushing the ball toward the rear.

But something's not quite right about this. There is nothing actually pushing the ball toward the rear, so its natural tendency should be NOT to accelerate. In fact, the plane is accelerating forward because of the thrust of the engines, and the tennis ball, free to roll, is falling behind. So, the force that appears to be acting on the tennis ball is fictitious. It's what physicists call a pseudo force. We observe pseudo forces whenever we ride in vehicles that are accelerating, decelerating, or taking a curved path.

The passengers also experience this pseudo force. We feel pushed back against our seats. Like the tennis ball, however, there's nothing actually pushing us back. Rather, our seats are pushing us forward so that we accelerate with the aircraft rather than falling behind. The real force is this forward push. If someone could stand by the side of the runway and look into the aircraft, he would see each passenger accelerating and the tennis ball going nowhere. He would make sense of this by saying that there's a real forward force on the passenger and none on the tennis ball. (Note to physicists: I'm neglecting the torque on the tennis ball.)

Takeoff from the viewpoint of a runway spectator

Let's redraw the picture to show the viewpoint of a flight attendant. From her viewpoint, the passenger is stationary and the tennis ball accelerates toward the rear. If she doesn't know that she is riding on an accelerating vehicle, she would need to explain the apparent acceleration of the ball by claiming the existence of a rearward force. She would also need to explain the stationarity of the passenger by claiming the existence of a rearward force that cancels the forward push the passenger feels on his back. These claimed forces are pseudo forces because there's no real physical thing, such as a spring, a magnet, or Arnold Schwarzenegger, creating them.

Takeoff from the viewpoint of a flight attendant

Centripetal force

Now let's look at a rider on the Mixing Bown ride from the viewpoint of someone looking down on the ride from a balcony.

Mixing Bowl rider from the viewpoint of a stationary observer

This observer sees the rider moving around a circle. He knows that Newton's Laws require a force to cause something to be deflected along a curve. If this force is discontinued, it will fly off on a straight path. For constant-speed motion around a circle, that force is toward the center of the circle. This is a real force; the rider feels it as pressure on her back. This force is called centripetal force.

Now suppose the rider can't look outside the bowl and doesn't know that the bowl is spinning. She looks to be stationary to herself and to her fellow riders. But she would still feel pressure on her back and would (correctly) say that this is a real force pushing her inward. The only way to explain her stationarity is to claim there's an outward force cancelling the inward force. This is a pseudo force because there's no Arnold pushing her outward. This pseudo force is called centrifugal force. It feels real to the riders because this is how our brains unconsciously make sense of the situation. The tendency of our bodies to move outward is actually caused by our bodies "trying" to fly off along a straight path.

Mixing Bowl rider from the viewpoint of other riders

It's OK to use pseudo forces to analyze motion

All this being said about pseudo forces being fictitious, we can still use them to predict the motions that the riders will observe within an accelerating or turning vehicle. In the airplane takeoff example, we could use the pseudo force on the tenns ball to predict its trajectory down the aisle and figure out how long it takes to reach the rear. We should just be aware that this analysis describes motion relative to the interior of the cabin.

In my discussions of tides and of the equatorial bulge, I take the viewpoint of a "rider" on the spinning and orbiting Earth and use centrifugal forces to explain things. This is because in those cases I believe this is more intuitive for the non-scientist.