Exploring Inertial Forces: What, Why & How

[fandi.bataineh]

it was mentioned in another thread that only inertial forces are frame dependent or relative.

what is an inertial force?
what are the differences between inertial and non-inertial forces?

and most importantly:
why inertial forces are frame dependent while other forces are not?

i tried wikipedia but they never give a clear picture there.
iam still a beginner at relativity so i hope to get full explanation here.

 

[WannabeNewton]

This isn't a relativity question. It's a purely mechanics based question. Inertial forces (sometimes called pseudo-forces) are defined as forces that arise in non-inertial frames. So for example centrifugal and Coriolis forces by definition only arise in rotating reference frames.They can always be transformed away, that is made to vanish, by boosting back to an inertial frame. Physical forces (e.g. tension in a string) on the other hand persist even in an inertial frame, which is the basic difference between inertial forces and physical forces. There's really nothing deeper to it than that.

Mathematically inertial forces arise in non-inertial frames such as rotating frames because ordinary time derivatives now have to be modified in order to take into account the acceleration and rotation of the spatial axes of the frame(s). These modifications are called "Christoffel symbols", which in this specific context encode the various time derivatives of the spatial axes of the frame(s). So for example in a rotating frame the centrifugal and Coriolis forces are both codified by an associated pair of Christoffel symbols.

EDIT: You're not going to get a full answer here because there are a lot of textbook problem sets you have to solve in order to get comfortable with doing mechanics in non-inertial frames using inertial forces. A simple forum reply does not constitute a substitute for that. I would recommend working through the problems in chapter 9 of Taylor "Classical Mechanics".

 

[HallsofIvy]

The simplest way to state it is that the "inertial forces" are, by definition, those that are frame dependent.

 

[pervect]

I'd say that forces transform as tensors only between two inertial frames, the notion of a generalized force in a generalized coordinate system isn't a tensor. But that's my personal conclusion, I haven't yet got a hold of a textook or paper that talks about the issue.

 

[A.T.]

what is an inertial force? what are the differences between inertial and non-inertial forces?

In simple terms:

Inertial forces appear in accelerated reference frames because of the acceleration of the frame, not because of the interaction between two objects.

Therefore:

- Inertial forces exist only in non-inertial reference frames.
- Inertial forces do not obey Newton's 3rd Law.

 

[willoughby]

Perhaps you don't understand what an inertial reference frame is. An inertial reference frame is any frame that is not accelerating at its most basic level. If you are moving with constant velocity in a straight line, that is an inertial reference frame. The importance of such frames is that the laws of physics apply in ALL inertial frames of reference. An example of this is you in a car driving straight down a highway at 60 mph. If you throw a ball up inside that car, the ball will fall back down to your hand just the same exact was as it would if you were standing still. The ball goes up, it comes straight back down. The fact that you are cruising along at 60 mph is irrelevant. Inside the car, you are in an inertial reference frame. If you blocked the windows, you would have no way of knowing if you were moving or how fast you were moving (assuming you couldn't feel road vibrations or hear road noise). Now imagine that you throw the ball up, and while the ball is in the air, the person driving slams on the brakes. What happens? The ball would go flying forward, right? How would you explain that within your frame? Relative to your frame of reference, the ball accelerated toward the front of the car, and if that's the case, then there must have been some force acting on the ball. Well, the truth is, there WAS no force. The ball just APPEARED to accelerate forward because your ENTIRE FRAME OF REFERENCE accelerated backward. While the brakes in the car were being applied, you were no longer in an inertial reference frame. Your reference frame was accelerating, and all APPARENT forces caused by the acceleration of the frame of reference are called inertial forces. As someone already pointed out, centrifugal and the coriolis forces are two examples of common inertial forces. They are called inertial forces because what actually caused the perceived force and acceleration of the object was the objects own inertia maintaining motion while the frame accelerated. (an object in motion remains in motion - inertia).

 

[noone00000]

They are called inertial forces because what actually caused the perceived force and acceleration of the object was the objects own inertia maintaining motion while the frame accelerated. (an object in motion remains in motion - inertia).

in general, is the direction of the inertial force in the direction of the original motion of the object, or in the direction of the original motion of the reference frame, or in the opposite direction of the acceleration of the reference frame?

lets say that in your previous example the ball was rolling in the car from side to side rather than being thrown upward, just to get rid of the effects of gravity to make things more clear. so in what direction would the inertial force on the ball would be?

 

[A.T.]

in the opposite direction of the acceleration of the reference frame?

Yes, for linearly accelerated reference frames:
http://en.wikipedia.org/wiki/Fictitious_force#Acceleration_in_a_straight_line

For rotating frames it's a bit more complex:
http://en.wikipedia.org/wiki/Centrifugal_force_(rotating_reference_frame)
http://en.wikipedia.org/wiki/Coriolis_force