(23) Accelerated Frames of Reference: Inertial Forces

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

F_centripetal = mV ²/R

To anyone who has not studied motions and accelerations, "centripetal" is probably a new word. However, most people are aware of the outwards-pushing 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 forces

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 roller-coaster 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 Vectors

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 y-axis 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.

The Decelerating Bus

Note that now, if equilibrium is to be maintained, the forces must change. The weight F₁ is fixed, and to keep the balance of the equation (as well as of the passenger), F₂ must change. For instance, the passenger may grab the seat in front, pushing her or his body back, with a force equal to the acceleration added on the other 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_upushing forward, in the direction of y_u. This 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 Off

If you ever take a trip on a jetliner, notice how during take-off 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 (a fishing sinker or a stack of screw-nuts does fine) and let it hang before take-off, defining the "down" direction. During take-off, the string will slant backwards, by perhaps 5-10 degrees.

All this is rather mild compared to the inertial forces felt by astronauts in the space shuttle during launch. The shuttle accelerates at about 2-3g, so the added force felt by the astronauts, in their frame of reference, is 2-3 times their weight.