
Note: This section uses vectors, and some way of denoting them on the board and in the notebook must be agreed on by the class. In this lesson plan, all vector quantities will be in boldface.

Index 22.Reference Frames 22a.Starlight Aberration 22b. Relativity 22c. Flight (1) 22d. Flight (2) 23. Inertial Forces 23a. The Centrifugal Force 23b. LooptheLoop 24a.The Rotating Earth 24b. Rotating Frames The Sun S1. Sunlight & Earth S1A. Weather S1B. Global Climate S2.Solar Layers 
The other case of a moving frame of reference studied here is an accelerated frame of reference.
An earlier section described motion in a circle and introduced the centripetal force which makes such motion possible. That is a force towards the center of rotation, of magnitude
To anyone who has not studied motions and accelerations, "centripetal" is probably a new word. However, most people are aware of the outwardspushing centrifugal force, the force which flings you away from the center of rotation in a car speeding around a corner. In a popular carnival ride, that is the force which pushes you against a rotating drum, holding you in place even when you are upside down. What is the connection?
Inertial ForcesAs will be seen, the centrifugal force is not a "real" force, in the sense that in any motion calculated "in the frame of the universe" (or in one moving uniformly with respect to the universe) it does not appear at all. In such a frame, if an object moves around a circle, a centripetal force is needed to maintain that motionotherwise it flies off at a tangent, with constant velocity along a straight line.

Unfortunately, if you sit (for instance) in a roller coaster car going around a vertical loop, as in the picture above, from an amusement park in Japan, it is a bit difficult to visualize your motion with respect to the fixed Earth. It is much simpler to orient yourself with respect to the car in which you sit. The rollercoaster car, however, is undergoing various accelerations, and as a rule, when we try to apply the laws of motion inside an accelerated frame of reference, extra forces enter the picture, known as inertial forces. You could call them "fictitious" forces, if you wish, because when the same motion is calculated in the frame of the outside world, they do not appear at all. Inside the accelerated frame of reference, however, they can't be told apart from real forces, and they need real forces to balance them. The centrifugal force is one such force, but before studying it, we consider a simpler case. You are sitting in a bus, moving in a straight line. Suddenly the driver applies brakes and all passengers feel pushed forward. Why? Unit VectorsForces are vectors, and F=ma is a vector equation. Rather than separate it into components, we introduce here a simple notation which allows vectors to be handled as single entities.Suppose we work in an (x,y) system of coordinates, with x pointing down and y pointing in the direction of the motion of the bus (if you are used to an (x,y) system with the yaxis pointing up, rotate it clockwise by 90°). Then unit vectors x_{u} and y_{u} would then be vectors of unit length, in the x and y directions.

[Note: marking unit vectors with a subscript "u" is a nonstandard notation, forced by the limitation of the HTML code used on the worldwide web. The standard notation is a caret on top, as in â, ê or û, but unfortunately, HTML does not allow carets to be placed on most letters. If presenting this material in a classroom, it may be better to use carets.] 
So, in (x,y) coordinates, a vector V with components (V_{x},V_{y}) could be written
The Decelerating BusLet us examine the forces on some passenger inside the bus; the x direction is down, y is in the direction in which the bus is moving. Two forces are involved: the weight F_{1}=mg x_{u}, pulling the passengers down, and the reaction F_{2} of the seat, which does not allow any motion in that direction. As long as the bus moves in a straight line and with a constant speed, these two are the only ones that matter and we get as condition of equilibrium
If equilibrium is to be maintained, the forces must change. The weight F_{1} is fixed, but to balance horizontal forces, the reaction force F_{2} must change. For instance, the passenger may grab the seat in front, pushing her or his body back, with a force equal to what the acceleration added on the right side of the equation. To see how the situation looks in the frame of the bus, we add +ma y_{u} to both sides. On the right now, what is added equals what is subtracted, leaving zero, so the equation becomes
This may be interpreted in the frame of the decelerating bus as follows. For forces to stay in equilibrium, all forces (as before) must add up to zero, but now they must include an inertial force may_{u} pushing forward, in the direction of y_{u}. Such an inertial force is only felt in the moving frame. You may call it a "fictitious force" if you wish. But when you need a seatbelt or an airbag to stop it from throwing you through a windshield, it seems real enough!
Taking OffIf you ever take a trip on a jetliner, notice how during takeoff you are pushed back in your seat. That is the inertial force acting in the frame of the accelerating airplane. In a decelerating bus, you are pushed forward, but in an accelerating airplane you are pushed back. For a simple experiment, take with you a weight on a string (e.g. a fishing sinker or a stack of screwnuts tied to the end of the string) and let it hang before takeoff, defining the "down" direction. During takeoff, the string will slant backwards, by perhaps 510 degrees.All this is rather mild compared to the inertial forces felt by astronauts on a rocket or on the space shuttle during launch. The rocket accelerates at about 23g, so the added force felt by the astronauts, in their frame of reference, is 23 times their weight.

Next Stop: #23a The Centrifugal Force
Timeline Glossary Back to the Master List
Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Updated: 9222004 ; reformatted 25 March 2006 ; later update 14 October 2016