Mechanics part 4: Galilean Relativity
Welcome back fellow carbon-based life forms! Last time, we talked about momentum and its conservation. As we saw, it allows us to analyze the interactions between particles just by knowing their initial momentum! Basically, if there are no external forces on a system, its momentum is always conserved. Always.
Today I want to talk about something different yet related. Reference frames. You may have heard of them before, in relativity and stuff. But what I’m gonna talk about today isn’t Einsteinian relativity, as they call it, but Galilean relativity. This form of relativity holds for object moving at speeds much, much slower than the speed of light. So yeah, no special or general relativity today.
Let’s start with the question in your head. What exactly is a reference frame? In simple words, a reference frame is a coordinate system attached to a body. In your reference frame, you measure the position and velocities of particles relative to you.
As you can see, the tree is 40 m away relative to A but 16 m away relative to B. So now let’s generalize it to multiple dimensions. We will have two frames S and S’, attached to A and B respectively. This means that we’ll have A positioned at O at all times, and B positioned at O’ at all times. We’ll call S the ‘ground frame’, and for our purposes is stationary. Let R be the position of B, and r be the position of point P relative to A, and r’ be the position of that same point relative to B.
As you can see,
Now we assume that B is moving with some non-zero velocity. Then the position of the P changes with time relative to B. Thus
Now let’s assume there’s an object C at point P, and it also moves relative to the ground, then
So as you can see above, the velocity of C relative to B is simply its velocity relative to A minus the velocity of B relative to A. Pretty neat huh? Now let’s differentiate again, this time assuming B moves with a constant velocity relative to A. We see that
So notice what we got. If B moves with a constant velocity, then the acceleration of C with respect to B is the same as with respect to A. Taking into consideration the fact that multiplying this result by the mass of particle C will give Newton’s 2nd law, we see that the force on particle C is the same relative to both A and B (because mass doesn’t — welp, that’s a story for another time)
This sort of reference frame, in which Newton’s laws remain the same relative to a stationary observer, is called an inertial frame. Basically the frame is moving at a constant velocity. Also, now’s a good time to point out that there is no such thing as absolute rest. This can be seen pretty easily. If you have a car moving at 50 km/hr relative to say, a tree, then the tree to moves at 50 km/hr in the opposite direction as the car, relative to the car. Absolute rest doesn’t exist.
So we just talked about inertial frames. But now it makes sense to ask, what if particle B is accelerating? How do Newton’s laws work out in that case? Turns out something very interesting pops up. Let’s do the math.
You see what’s interesting? We get an extra term in this expression for acceleration relative to B, and this term is the acceleration of B itself. Now multiplying by the mass of C,
Here, F’ is the force on C in B’s frame, F is the force in A’s frame, and A is the acceleration of B. We have discovered something fascinating! In an accelerating (non inertial) reference frame, a particle feels not only the forces acting on it in an inertial frame, but also a ‘fictitious’ force which points in the opposite direction of the frame’s acceleration! This is why when a car speeds up, you feel as if something is pulling you to your seat. Relative to the ground, there are no forces acting on you (the thrust is on the car), so F is zero. Assuming you have mass m and the car accelerates at a, the force pulling you to your seat is F=-ma, which points opposite to the direction of the car’s acceleration!
One last point. You may be wondering why these forces are termed as ‘fictitious’. Well, this is because a force is said to be ‘real’ only if it exists in all reference frames, ie observers in all reference frames agree that the body experiences the force. In the case of non inertial frames, the pseudo force (as its also called) is only felt by an observer in that frame, and not an observer in an inertial frame. In fact, according to general relativity, gravity too is such a force!And according to special relativity, the magnetic force is a pseudoforce too! Another thing I didn’t mention was that distances between places and the time between events stays the same in all reference frames. This isn’t special relativity, after all. Lastly, all inertial frames are equally valid, ie if an object is moving in one frame, it may not be moving in another, but the laws of physics remain the same. There is no special inertial frame.
So that’s it for today! Now you know about reference frames, and how motion can cause observers to, well, observe the motion of other particles differently! See ya!
Note: I am a high schooler and this article is based on my understanding of physics. Feel free to correct me! 😅