118 |
Week 2: Newton’s Laws: Continued |
2.3.1: Time
Let us start by thinking about time. Suppose that you wish to time a race (as physicists). The first thing one has to do is understand what the conditions are for the start of the race and the end of the race. The start of the race is the instant in time that the gun goes o and the racers (as particles) start accelerating towards the finish line. This time is concrete, an actual event that you can “instantly” observe64. Similarly, the end of the race is the instant in time that the racers (as particles) cross the finish line.
Consider three observers timing the same racer. One uses a “perfect” stop watch, one that is triggered by the gun and stopped by the racer crossing the finish line. The race starts at time t = 0 on the stop watch, and stops at time tf , the time it took the racer to complete the race.
The second doesn’t have a stop watch – she has to use their watch set to local time. When the gun goes o she records t0, the time her watch reads at the start of the race. When the racer crosses the finish line, she records t1, the finish time in local time coordinates. To find the time of the race, she converts her watch’s time to seconds and subtracts to obtain tf = t1 − t0, which must non-relativistically65 agree with the first observer.
The third has just arrived from India, and hasn’t had time to reset his watch. He records t′0 for the start, t′1 for the finish, and subtracts to once again obtain tf = t′1 − t′0.
All three of these times must agree because clearly the time required for the racer to cross the finish line has nothing to do with the observers. We could use any clock we wished, set to any time zone or started so that “t = 0” occurs at any time you like to time the race as long as it records times in seconds accurately. In physics we express this invariance by stating that we can change clocks at will when considering a particular problem by means of the transformation:
t′ = t − t0 |
(192) |
where t0 is the time in our old time-coordinate frame that we wish to be zero in our new, primed frame. This is basically a linear change of variables, a so-called “u-substitution” in calculus, but because we shift the “zero” of our clock in all cases by a constant, it is true that:
dt′ = dt |
(193) |
so di erentiation by t′ is identical to di erentiation by t and:
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d2~x |
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d2~x |
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F = m |
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= m |
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(194) |
dt2 |
dt′2 |
That is, Newton’s second law is invariant under uniform translations of time, so we can start our clocks whenever we wish and still accurately describe all motion relative to that time.
2.3.2: Space
We can reason the same way about space. If we want to measure the distance between two points on a line, we can do so by putting the zero on our meter stick at the first and reading o the distance of the second, or we can put the first at an arbitrary point, record the position of the second, and subtract to get the same distance. In fact, we can place the origin of our coordinate system anywhere we like and measure all of our locations relative to this origin, just as we can choose to start our clock at any time and measure all times relative to that time.
64For the purpose of this example we will ignore things like the speed of sound or the speed of light and assume that our observation of the gun going o is instantaneous.
65Students not going on in physics should just ignore this adverb. Students going on in physics should be aware
that the real, relativistic Universe those times might not agree.
Week 2: Newton’s Laws: Continued |
119 |
Displacing the origin is described by: |
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~x′ = ~x − ~x0 |
(195) |
and as above, |
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d~x′ = d~x |
(196) |
so di erentiating by ~x is the same as di erentiating by a displaced ~x′.
However, there is another freedom we have in coordinate transformations. Suppose you are driving a car at a uniform speed down the highway. Inside the car is a fly, flying from the back of the car to the front of the car. To you, the fly is moving slowly – in fact, if you place a coordinate frame inside the car, you can describe the fly’s position and velocity relative to that coordinate frame very easily. To an observer on the ground, however, the fly is flying by at the speed of the car plus the speed of the car. The observer on the ground can use a coordinate frame on the ground and can also describe the position of the fly perfectly well.
S’ |
S |
x’ |
x |
vt |
Figure 26: The frame S can be thought of as a coordinate system describing positions relative to the ground, or laboratory, “at rest”. The frame S′ can be thought of as the coordinate system inside (say) the car moving at a constant velocity ~v relative to the coordinate system on the ground. The position of a fly in the ground coordinate frame is the position of the car in the ground frame plus the position of the fly in the coordinate frame inside the car. The position of the fly in the car’s frame is the position of the fly in the ground frame minus the position of the car in the ground frame. This is an easy mental model to use to understand frame transformations.
In figure 26 one can consider the frame S to be the “ground” frame. ~x is the position of the fly relative to the ground. The frame S′ is the car, moving at a constant velocity ~v relative to the ground, and ~x′ is the position of the fly relative to the car. Repeat the following ritual expression (and meditate) until it makes sense forwards and backwards:
The position of the fly in the coordinate frame of the ground is the position of the fly in the coordinate frame of the car, plus the position of the car in the coordinate frame of the ground.
In this way we can relate the position of the fly in time in either one of the two frames to its position in the other, as (looking at the triangle of vectors in figure 26):
~x(t) |
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~x′(t) + ~vframet |
or |
~x′(t) |
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~x(t) − ~vframet |
(197) |
We call the transformation of coordinates in equations 197 from one (inertial) reference frame to another moving at a constant velocity relative to the first the Galilean transformation. Note that we use the fact that the displacement of the origin of the two frames is ~vt, the velocity of the
120 |
Week 2: Newton’s Laws: Continued |
moving frame times time. In a bit we’ll show that this is formally correct, but you probably already understand this pretty well based on your experiences driving cars and the like.
So much for description; what about dynamics? If we di erentiate this equation twice, we get:
d~x |
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d~x′ |
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+ ~vframe |
(198) |
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dt |
dt |
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d2~x |
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(199) |
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dt2 |
dt2 |
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(where we use the fact that the velocity ~vframe is a constant so that it disappears from the second derivative) so that if we multiply both sides by m we prove:
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d2~x |
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d2~x′ |
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F = m |
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= m |
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(200) |
dt2 |
dt2 |
or Newton’s second law is invariant under the Galilean transformation represented by equation 197 – the force acting on the mass is the same in both frames, the acceleration is the same in both frames, the mass itself is the same in both frames, and so the motion is the same except that the translation of the S′ frame itself has to be added to the trajectory in the S frame to get the trajectory in the S′ frame. It makes sense!
Any coordinate frame travelling at a constant velocity (in which Newton’s first law will thus apparently hold66) is called an inertial reference frame, and since our law of dynamics is invariant with respect to changes of inertial frame (as long as the force law itself is), we have complete freedom to choose the one that is the most convenient.
The physics of the fly relative to (expressed in) the coordinate frame in the car are identical to the physics of the fly relative to (expressed in) the coordinate frame on the ground when we account for all of the physical forces (in either frame) that act on the fly.
Equation 197, di erentiated with respect to time, can be written as:
~v′ = ~v − ~vframe |
(201) |
which you can think of as the velocity relative to the ground is the velocity in the frame plus the velocity of the frame. This is the conceptual rule for velocity transformations: The fly may be moving only at 1 meter per second in the car, but if the car is travelling at 19 meters per second relative to the ground in the same direction, the fly is travelling at 20 meters per second relative to the ground!
The Galilean transformation isn’t the only possible way to relate frames, and in fact it doesn’t correctly describe nature. A di erent transformation called the Lorentz transformation from the theory of relativity works much better, where both length intervals and time intervals change when changing inertial reference frames. However, describing and deriving relativistic transformations (and the postulates that lead us to consider them in the first place) is beyond the scope of this course, and they are not terribly important in the classical regime where things move at speeds much less than that of light.
66This is a rather subtle point, as my colleague Ronen Plesser pointed out to me. If velocity itself is always defined relative to and measured within some frame, then “constant velocity” relative to what frame? The Universe doesn’t come with a neatly labelled Universal inertial reference frame – or perhaps it does, the frame where the blackbody background radiation leftover from the big bang is isotropic – but even if it does the answer is “relative to another inertial reference frame” which begs the question, a very bad thing to do when constructing a consistent physical theory. To avoid this, an inertial reference frame may be defined to be “any frame where Newton’s First Law is true, that is, a frame where objects at rest remain at rest and objects in motion remain in uniform motion unless acted on by an actual external force.
Week 2: Newton’s Laws: Continued |
121 |
2.4: Non-Inertial Reference Frames – Pseudoforces
Note that if the frame S′ is not travelling at a constant velocity and we di erentiate equation 201, one more time with respect to time then:
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dt |
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dt |
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d2~x |
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d2~x′ |
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+ ~aframe |
(202) |
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dt2 |
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dt2 |
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or |
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~a = ~a′ + ~aframe |
(203) |
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Note that the velocity transformation is unchanged from that in an inertial frame – the velocity of the fly relative to the ground is always the velocity of the fly in the car plus the velocity of the car, even if the car is accelerating.
However, the acceleration transformation is now di erent – to find the acceleration of an object (e.g. a fly in a car) in the lab/ground frame S we have to add the acceleration of the accelerating frame (the car) in S to the acceleration of in the accelerating frame S’.
Newton’s second law is then not invariant. If S is an inertial frame where Newton’s Second Law is true, then:
~ |
′ |
(204) |
F |
= m~a = m~a + m~aframe |
We would like to be able to write something that looks like Newton’s Second Law in this frame that can also be solved like Newton’s Second Law in the (accelerating) frame coordinates. That is, we would like to write:
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F~ ′ = m~a′ |
(205) |
If we compare these last two equations, we see that this is possible if and only if: |
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~ ′ |
~ |
~ ~ |
(206) |
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= F − m~aframe = F − F p |
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~
where F p is a pseudoforce – a force that does not exist as a force or force rule of nature – that arises within the accelerated frame from the acceleration of the frame.
In the case of uniform frame accelerations, this pseudoforce is proportional to the mass times a the constant acceleration of the frame and behaves a lot like the only force rule we have so far which produces uniform forces proportional to the mass – gravity near Earth’s surface! Indeed, it feels to our senses like gravity has been modified if we ride along in an accelerating frame – made weaker, stronger, changing its direction. However, our algebra above shows that a pseudoforce behaves consistently like that – we can actually solve equations of motion in the accelerating frame using the additional “force rule” of the pseudoforce and we’ll get the right answers within the frame and, when we add the coordinates in the frame to the ground/inertial frame coordinates of the frame, in those coordinates as well.
Pseudoforces are forces which aren’t really there. Why, then, you might well ask, do we deal with them? From the previous paragraph you should be able to see the answer: because it is psychologically and occasionally computationally useful to do so. Psychologically because they describe what we experience in such a frame; computationally because we live in a non-inertial frame (the surface of the rotating earth) and for certain problems it is the solution in the natural coordinates of this non-inertial frame that matters.
We have encountered a few pseudoforces already, either in the course or in our life experience. We will encounter more in the weeks to come. Here is a short list of places where one experiences pseudoforces, or might find the concept itself useful in the mathematical description of motion in an accelerating frame: