Download Inertial and Non-inertial Reference Frames

Inertial and Non-inertial Reference Frames
Imagine a ball dropped from the roof of a building. It falls straight down with an acceleration of
9.8 m/s2. Now consider a ball dropped from an airplane in level flight with a constant velocity.
To an observer stationary on the ground, the ball appears to follow a parabolic path. However,
according to an observer moving with the airplane, it appears to drop straight down with an
acceleration of 9.8 m/s2 (provided there is no air resistance). In both cases, the observer could
analyze the motion using Newton's laws, in particular, the 2 nd Law. The only difference is that in
one frame of reference the ball has a horizontal component to its velocity, and in the other (the
airplane’s frame) it doesn't. In both frames the observers would reach the same conclusion: the
vertical motion can be explained by a constant vertical force given by F  mg .
This forms the basis of what is known as the Galilean Relativity Principle
Galilean Relativity Principle
The laws of mechanics are the same in reference frames which are moving
relative to each other with constant velocity.
In other words, we should be able to use Newton's laws to correctly describe any events, as
long as we use a reference frame moving at a constant velocity.
Accelerating Reference Frames
On a moving train, we can choose to use the train as our frame of reference. Imagine a coffee
cup sitting on a table in the train. As long as the train is traveling with a constant velocity, the
cup just sits on the table. Its motion can easily be described in either the reference frame of the
train, or in the reference frame of an observer standing still on the ground. In both reference
frames, the forces on the cup are balanced.
However, if the train slows down suddenly, or goes around a sharp corner, the cup can slide
along the table and fall off (if there is no friction). In the Earth's reference frame this is easily
explained by Newton's 1st law: the cup continues its original motion since there is no
unbalanced force acting to stop it. However, in the reference frame attached to the train
(which accelerates with the train) the cup would appear to slide forward or backward or
sideways, all on its own, without any apparent application of force.
This can't be explained using Newton's Laws in the train's frame since there was no force
exerted on the cup and therefore it shouldn't have accelerated. If we insisted on using
Newton's Laws, we would have to invent some sort of force that would be responsible for the
cup's motion. The problem with this "force" is that it doesn't exist – it is an artifact of the
reference frame we used. This is referred to as a fictitious force.
The diagrams below illustrate how two different observers would explain this event if the train
was moving to the right and slowing down.
Earth Frame
FN
The velocity of the cup is
constant as there is no
unbalanced force acting
on it.
FN
Since the cup
accelerates forward (to
the right), there acc.
must
be some unbalanced
force in that direction.
Fg
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Train Frame
F?
Fg
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We can tell if a "force" is fictitious or not by asking whether or not it makes sense when
discussed within the context of the 3rd Law. According to the 3rd law, every force is actually a
two-way interaction. In the example above, if a force does truly push on the cup, then the cup
must also push back on something else (whatever is causing the cup to accelerate). However,
since there are no new objects around to act as the source of the force, we have to conclude
that the force is an illusion – caused by our choice of an accelerated reference frame. This
means that Newton's laws of motion don't apply in accelerated reference frames.
The idea of an accelerated reference frame also helps explain why we seem to experience an
outward force if we are on a merry-go-round or going around a corner in a car. When we are in
a moving vehicle (or on a rotating platform) we tend to use that vehicle as our mental reference
frame so from our perspective, we are not moving within that frame. If the vehicle goes around
a corner, or if the platform rotates, our reference frame is accelerating (changing direction). In
this type of situation, we often feel like we are going to be “flung off” away from the centre of
the curve or platform. To prevent ourselves from being flung off in the case of a merry-goround, we tend to hang onto something to pull ourselves inward against the outward “force”.
As a result, we think there is some force pushing us away from the centre of the circle (often
called the centrifugal force). However, this “force” is an illusion. If we try to apply the 3rd law, it
says that we should also be pushing inward on whatever is producing the outward force.
However, no matter how hard we try, we won't find that object because it doesn't exist. In
reality, what we are experiencing is an inward centripetal force. In a car this would usually be
provided by friction from the seat, or possibly from the normal force due to contact with a door
if we are leaning against it. On a merry-go-round, that force could be supplied by a pole that we
hang on to, or friction between our feet and the platform.
The two different points of view for explaining the forces when we are on a merry-go-round are
illustrated below. In this case the person is holding on to a pole.
Earth Frame
Merry-go-Round Frame
Fout?
I am going in a circle so
there must be an
unbalanced force
acting on me. I can feel
the pole pulling on my
hand so it must be
providing that force.
acc.
Fpole
I am not going anywhere
relative to the platform so
the forces on me must be
balanced. But, I can feel
the pole pulling on my
hand. That means there
must be another force
acting on me in the
opposite direction.
Fpole
The person using the merry-go-round frame now has the difficulty of explaining what they are
pushing inward on if they believe that something is pushing out on them.
These examples show that we can only apply Newton's Laws of motion in certain reference
frames. Reference frames in which Newton's laws of motion do work are called inertial
reference frames (IRF's). IRF's have certain properties;
1. Inertial reference frames move with constant velocity.
2. Newton's laws hold only in inertial reference frames.
3. Newton's laws hold in all inertial reference frames.
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It has been suggested that we can use this knowledge to simulate (not create) the effects of
gravity in a space station. If the space station is a large, rotating cylinder or wheel, people on
the inner surface of the cylinder and who are moving with it will be in an accelerated frame
(since they are travelling in circles). In this case, the force that is responsible for making them
go in a circle is the normal force due to the contact between their feet and the wall of the
cylinder. At the right speed, that contact force would be exactly the same size as the force of
gravity would be if the person was on Earth. The person would feel as if they were standing
upright on a level surface.
Space Station:
Outside Frame, at rest
Inside station frame (moving)
Fout? = Fg?
acc.
no acc.
FN
FN
dropped apple will land at feet
This simulated “gravity” will also affect objects that are released. An apple moving with a
person will continue in a straight line when it is released (thanks to the 1 st Law). It will
eventually hit the “floor” but in that time the person’s feet will also have moved so that the
feet meet the apple, making it appear as if the apple has dropped to their feet. This is only true
if the apple doesn’t fall too far. If it is dropped from a great height, the fact that the
acceleration varies with distance from the centre means the motion of the apple will not
appear to have a constant acceleration and its “fall” will not be straight “down”.
Note: the Earth's surface is not an IRF since the Earth is
rotating. Since the acceleration produced by the
rotation of the Earth is quite small, we don't normally
have problems assuming we are in an IRF. However,
when objects are traveling large distances over the
surface of the Earth, the fact that it is not an IRF can
lead to the introduction of fictitious forces. An example
is the fictitious force known as the coriolis force which
is often used to explain weather phenomena. It is not
really a force, but rather an apparent force which
arises due to the Earth's spin and shape. A better term
is the coriolis effect.
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Accelerating frames can not only introduce new fictitious forces, they can also lead to the
apparent elimination of real forces. Inside a large jet plane that is accelerating downward at
with the same acceleration as gravity, passengers seem to float in mid-air. They appear to be
weightless, but only because their reference frame is accelerating. The force of gravity from the
Earth on them is very real and it is still there. However, their floating behaviour suggests two
possibilities – either gravity disappears, or
some new force cancels gravity, neither of
which is true. Thus, weightlessness can be
simulated by an airplane flying on a parabolic
arc in such a way that it is always accelerating
down at 9.8 m/s2. This is how astronauts can
train for space flight.
Note that astronauts in orbit around Earth are also not weightless. The force of gravity is still
significant in orbit. The astronauts merely appear weightless since we are observing them with
reference to an accelerating frame (the shuttle or space station).
Another device that makes use of spinning to simulate gravity is the centrifuge. These are often
used to spin test tubes with fluids in them at very high speeds so that the components of the
fluid separate according to density. Gravity will normally do this, but it can take a long time. By
simulating a higher gravity, the separation is quicker.
The acceleration of a moving reference frame can be measured using a simple accelerometer
which is just mass hanging on a string. In an accelerating vehicle, the hanging will hand at some
angle opposite the direction of acceleration.
Ground Frame
acc.
FT
θ
Accelerated Frame
FNet = 0
FT
θ
F?
FNet
Fg
x: ma = FT sin(θ)
y: 0 = FT cos(θ) – Fg
 FT = mg/ cos(θ)
ma = (mg/ cos(θ))sin(θ) = mg tan(θ)

a = g tan(θ)
.
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Fg
In this frame, it appears
as if some backwards
force pushes on the
hanging weight.
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