#23 Accelerated Frames of Reference: Inertial Forces
#23a Frames of Reference: The Centrifugal ForceUniformly moving frames of reference experience no new forces. Uniformly accelerating frames and rotating frames do so. If we want to express the equations of motion in their own frame of reference, we must always add "inertial forces" to represent the effects of their acceleration. The centrifugal force is one such force, described here and illustrated by examples. 
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements:
#23 "Accelerated Frames of Reference: Inertial Forces" Sframes2.htm,
#23a "Frames of Reference: The Centrifugal Force" Sframes3.htm,
Related web pages (not used in the lesson)
"From Stargazers to Starships" home page: ....stargaze/Sintro.htm 
Goals: The student will learn
Stories: Variation of g and the bulge of the Earth. (Optional, at the end of this lesson plan: pumping a swing)
Starting the lesson: Up to now we have dealt with uniformly moving frames of reference. That was easy: moving from one such frame to another, it was only necessary to redefine velocities and coordinates in terms appropriate to the new frame. The forces were the same, because forces, as Newton showed, depend only on acceleration. Tody we go a step further, to accelerating frames of referencein particular, two examples:
From experience everyone knows what to expect. In the first case, the passengers are flung forward, in the second, they are pushed to the outside of the curve, by what is known as a centrifugal force. Both these are a rather special type of force, called inertial forces: they only appear in the accelerating frame.
For example: the bus which suddenly stops.
This can be quite confusing. Some teachers and textbooks avoid inertial forces altogether or call them "fictional forces". As we will see, however, they can be quite useful. One thing you should remember: be sure you know to which frame of reference your coordinates belong. With accelerated frames, all frames are not all equivalent. *********** The best known example, of course, is a rotating frame of reference, where the centripetal force pushes inwards, and the centrifugal force pushes outwards. Which to use? Here is the rule: [on the boardall please copy]
The centrifugal force is only used in a rotating frame. It must be added to other forces in that frame, in order to take into account the frame's rotation, and is directed away from the axis of rotation. As you will see, both describe the same physics, and either can be used to study motion. However, the centrifugal force seems more intuitive: this is what you feel when you sit in a car going around a sharp curve. You feel yourself pushed to the outside of the curve. Your mind is keyed to the frame of reference of the car that surrounds you, and in that frame you feel pushed outwards. As a result, almost anyone knows intuitively what a centrifugal force is, while "centripetal force" is only familiar if you learned about it in school. (Then continue with the material of section #23 in "Stargazers.")
Teaser Ask students if anyone knows what is a "Thankyouma'am"that is, as a nounphrase, name of some object. And how is it connected to inertial forces? (If no one does, challenge the class to find out by the time of the next lesson, cautioning them that a pretty detailed dictionary may be needed.)
A "Thankyouma'am" (=madam) is a sharp turn in the road, or a sharp angle at the bottom of a dip in the road. It's a humorous American expression, probably from the stagecoach travel of the 1800s. When a stagecoach went through such a sudden turn, unprepared passengers were thrown against each othernot exactly pleasant, if you had hefty neighbors! The culprit, of course, was the inertial force caused (in the frame of the stagecoach) by the sudden turn. An astronaut lies horizontally on a couch, flat on the back, inside a space shuttle as it takes off. In which direction is the astronaut accelerating?
Does the astronaut feel pushedand if so, which way?
We agree the astronaut's body is pressed towards the couch. Viewed from the outside world, why that pressure?
Viewed from inside the cabin, why that pressure?
When working with vectors in cartesian coordinates (x,y), it is useful to define unit vectors x_{u} and y_{u}. These are vectors of unit length in the directions of the coordinate axes
What are unit vectors useful for?
Say you fire a gun at an angle upwards, so that
Let the direction of u be the y axis, and of w the x axis. After t seconds, the upward velocity is (u–gt) while the horizontal one remains w. Express the velocity vector V after time t.
Is motion around a circle with constant speed ("uniform rotation") an accelerated motion?
Why is it accelerated, if the speed does not change?
So, for a body with mass m to move with constant speed V around a circle of radius R, a force is needed. What is that force called, what is its magnitude and what is its direction?
You are standing on the floor, and your body is subject to two forces:
[The discussion that follow may be helped by a crude drawing on the board].
You are sit on a whirling turntable, part of a carnival ride, at a distance R from the axis, going around at a velocity V. As before, F_{1} is your weight and F_{2} is the force exerted on you by the ride.
or The total force F_{1} + F_{2} on youin the first case it is outwards, in the second inwards. Which is it?
Why?
But wait! Another way of writing the same equation is
Note: the subject below is discussed in more detail in #24a "The Rotating Earth". You live on a rotating Earth. How does the centrifugal force affect you on the equator?
How does the centrifugal force affect you away from the equator?
In which direction is the shift?
Falling bodies are slightly displaced towards the equator.
Does the centrifugal force of the Earth's rotation affect the Earth itself?
Originally the meter was defined in terms of the distance from the equator to the pole, which was to be equal to 10,000,000 meters. If that were still the definition,would you expect the length of the equator to be 40,000 km? A little more? A little less?
Note: the actual meter was defined as the distance between two fine scratches on a "standard meter" metal bar kept in Paris, supposedly based on the above distance. Later it was redefined in terms of wavelengths of light, which can be very precisely measured.
Jupiter and Saturn both rotate around their axes in about 10 hours. Would you expect them to be more oval or more spherical than the Earth?
[Optional] You are sitting on a rotating platform, right on the middle. You want one of the people sitting nearer to the edge of the platform to join you at the middle, so you extend your hand and pull that person in, against the centrifugal force. What do you think: do you need energy to overcome the centrifugal force? Does your arm perform work?
There exists an interesting application. You are probably aware children can "pump up" a swing and keep it going; you may even have done so yourself. How is it done? The process is best seen in swings on which you stand rather than sit. The centrifugal force on the swing varies: it is greatest at the bottom of the swing, where motion is fastest, but zero when the swing briefly stops at the end points. What you do is, near the end points you lower your body, then near the bottom of the swing you stand up straight again. By standing up, you overcome the centrifugal force, which is directed towards your legs, so you invest energy. But energy is conserved and must go to somewhere else in the motion! It actually goes to the energy of the swinging motion, making the swing move more vigorously, or at least making it overcome the energy loss to friction, which would otherwise gradually slow it down. The "pumping" of a swing by moving legs and body while sitting down is somewhat similar.

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