Acceleration due to fictitious force independent of mass?

[Frank Castle]

I have been asked by someone if it is true that in general, for a constantly accelerating reference frame, i.e. a non-inertial reference frame, the acceleration of a particle (as observed in this frame) due to the corresponding fictitious force is independent of its mass.

My response was yes. My reasoning being that the apparent acceleration of any object within the accelerating frame of reference is due to the fact that the frame itself is accelerating, and not the objects themselves (neglecting any other external forces that may be acting on them). As such, relative to this non-inertial reference frame, all objects within the frame will appear to accelerate in the opposite direction at the same rate. For example, if a car is accelerating to the rate at a constant rate, an observer in the car would observe objects at the front of the car accelerate towards the back of the car at exactly the same rate, independently of their respective mass, such that they will hit the back of the car at exactly the same time (ignoring any air resistance).

I then proceeded to show this mathematically, by noting from Newton's 2nd law, that the acceleration of an object is proportional to the applied force. I iterated that Newton's 2nd law is not a definition of force, i.e. in general ##f\neq ma## (for example, Coulomb's law states that ##f=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}##, and this mathematical expression defines the force acting on a charge due to the presence of another charge), but it is the statement that the rate of change of momentum is equal to the applied force in a special set of reference frames, so-called inertial frames.
However, in this case, the apparent force acting on the particles in the non-inertial reference frame (in the absence of any actual external forces), is an artifact of the acceleration of the reference frame itself, and hence given that it can be shown, that in general for a constantly accelerating reference frame, $$\mathbf{f}'=\mathbf{f}+m\mathbf{a}_{0}$$ where ##\mathbf{a}_{0}## is the acceleration of the non-inertial reference frame, ##\mathbf{f}'=m\mathbf{a}'## are the net forces acting on the particles as observed in this frame, and ##\mathbf{f}## are the net forces acting on the particles as observed from an inertial frame.
From this we can define the fictitious force, ##\mathbf{f}_{fict}## as $$\mathbf{f}_{fict}=-m\mathbf{a}_{0}$$ hence we see that if there are no net external forces acting on the particles, then relative to the non-inertial reference frame we will have that $$-m\mathbf{a}_{0}=\mathbf{f}_{fict}=\mathbf{f}'=m\mathbf{a}'\Rightarrow \mathbf{a}'=-\mathbf{a}_{0}$$ and so, relative to an observer in the non-inertial reference frame, all particles will appear to accelerate in the opposite direction to the acceleration of the frame, independently of their mass.

I think I've correctly informed them, but I'm now doubting myself. I don't want to convey incorrect information and would appreciate someone taking a look at this and letting me know what they think (importantly, letting me know if anything is incorrect about it).

 

[drvrm]

the frame itself is accelerating, and not the objects themselves

the frame and the objects are also accelerating.

I iterated that Newton's 2nd law is not a definition of force,

Newtons second law does give a definition of force in classical mechanics.
in different field of force the causes may not be mechanical but if 'motion' is classical Newtons's laws hold.

the apparent force acting on the particles in the non-inertial reference frame (in the absence of any actual external forces), is an artifact of the acceleration of the reference frame itself,

by artefact one means an artificial construct - i beg to differ the pseudo forces are very much real.

However, in this case, the apparent force acting on the particles in the non-inertial reference frame (in the absence of any actual external forces), is an artifact of the acceleration of the reference frame itself, and hence given that it can be shown, that in general for a constantly accelerating reference frame,

f′=f+ma0f′=f+ma0​

\mathbf{f}'=\mathbf{f}+m\mathbf{a}_{0} where a0a0\mathbf{a}_{0} is the acceleration of the non-inertial reference frame, f′=ma′f′=ma′\mathbf{f}'=m\mathbf{a}' are the net forces acting on the particles as observed in this frame, and ff\mathbf{f} are the net forces acting on the particles as observed from an inertial frame.
From this we can define the fictitious force, ffictffict\mathbf{f}_{fict} as

ffict=−ma0ffict=−ma0​

\mathbf{f}_{fict}=-m\mathbf{a}_{0} hence we see that if there are no net external forces acting on the particles, then relative to the non-inertial reference frame we will have that

−ma0=ffict=f′=ma′⇒a′=−a0−ma0=ffict=f′=ma′⇒a′=−a0​

-m\mathbf{a}_{0}=\mathbf{f}_{fict}=\mathbf{f}'=m\mathbf{a}'\Rightarrow \mathbf{a}'=-\mathbf{a}_{0} and so, relative to an observer in the non-inertial reference frame, all particles will appear to accelerate in the opposite direction to the acceleration of the frame, independently of their mass.

well i could not follow your arguments as you were taking the frame acceleration as a(0) and again as a' and by assumtion it was equal but you proved the equality -i do not know why you did that.
its better to stay in a lift which can be accelerated and a spring balance with a load -then the equations/relations will be personally satisfactory.
well i may have been confused -that i can admit.

 

[A.T.]

I have been asked by someone if it is true that in general, for a constantly accelerating reference frame, i.e. a non-inertial reference frame, the acceleration of a particle (as observed in this frame) due to the corresponding fictitious force is independent of its mass.

This is actually true for all inertial (fictitious) forces, not just for the one in a constantly accelerating reference frame.

 

[Frank Castle]

the frame and the objects are also accelerating.

The objects do not experience an acceleration relative to an inertial observer until they are subjected to a contact force, for example, when you are accelerating in a car the reason you feel a force is because the contact force of the car seat is pushing on you causing you to accelerate with the frame, i.e. the car. But objects on the dashboard, with nothing hindering them, will appear to accelerate towards the back of the car, all at the same rate independent of mass, until they reach an obstacle such as the back seat, at which point they will be accelerated forwards with the car due to the contact force of the seat acting on it. The point being that relative to the observer in the accelerating frame of the car, there appears to be a force acting on the objects on the dashboard causing them to all accelerate at the same rate and hence appears to be proportional to their mass. However, to an observer in an inertial reference frame, where Newton's laws hold, it is clear that the objects are not actually accelerating (until they hit the back seat), but the are at rest. It is the car that is accelerating around them which makes it appear as though they are accelerating in the reference frame of the car.

Newtons second law does give a definition of force in classical mechanics.
in different field of force the causes may not be mechanical but if 'motion' is classical Newtons's laws hold.

Newton's 2nd law defines a relationship between a force and the resulting acceleration of the object it is acting on. This relationship holds in all inertial frames, but it does not define what the force is and how it acts, this has to be determined empirically (or be postulated and tested experimentally). For example, Hooke's law states that the force required to stretch or compress a spring is proportional to the displacement of the spring from it's equilibrium position, i.e. ##\mathbf{f}=k\mathbf{x}##, and according to Newton's 2nd law, this is proportional to the corresponding acceleration of the spring, i.e. ##k\mathbf{x}=m\mathbf{a}##.

by artefact one means an artificial construct - i beg to differ the pseudo forces are very much real.

It is an artificial construct to account for the apparent forces acting on objects due to acceleration of the reference frame. It is fictitious, the actual forces felt are due to contact forces (which are real forces) which act to accelerate the objects along with the frame (c.f. my car example earlier in this post)

frame acceleration as a(0) and again as a'

No, ##\mathbf{a}_{0}## is the acceleration of the reference frame and ##\mathbf{a}'## is the acceleration of the objects in the accelerating reference frame relative to an observer in that frame.
Although I do admit my "proof" is somewhat tautological.

 

[Frank Castle]

This is actually true for all inertial (fictitious) forces, not just for the one in a constantly accelerating reference frame.

Good point!
Is what I put correct apart from that?

 

[drvrm]

It is an artificial construct to account for the apparent forces acting on objects due to acceleration of the reference frame. It is fictitious, the actual forces felt are due to contact forces (which are real forces) which act to accelerate the objects along with the frame (c.f. my car example earlier in this post)

the name fictitious does give a flavor of artificial but for an observer say on Earth the centrifugal forces and coriolis forces do act -no doubt to an inertial observer this looks weired . or a man moving on the death-well and his motorcycle defying gravity does experience the centrifugal force.
the centrifuge machines uses the centrifugal acceleration -say washing machines.
so for me its hard to think as 'fictitious'.

 

[A.T.]

so for me its hard to think as 'fictitious'.

Then call them "inertial forces", and stop obsessing about a naming convention.

 

[Frank Castle]

the name fictitious does give a flavor of artificial but for an observer say on Earth the centrifugal forces and coriolis forces do act -no doubt to an inertial observer this looks weired . or a man moving on the death-well and his motorcycle defying gravity does experience the centrifugal force.
the centrifuge machines uses the centrifugal acceleration -say washing machines.
so for me its hard to think as 'fictitious'.

These are all fictitious (inertial) forces. The centrifugal and coriolis forces arise because the Earth is an accelerating reference frame. The only real force that a man on a motorcycle in the death well experiences is the centripetal force (again, the centrifugal force is an inertial force that appears because he is in a non-inertial reference frame).

 

[hackhard]

I think I've correctly informed them, but I'm now doubting myself.

you have complicated an obvious concept
suppose B is at rest is frame A,
frame C moves x(C) displacement in frame A
so B moves -x(C) displacement in frame C
differentiate over time = -v(C)
differentiate over time = -a(C)

 

[Frank Castle]

you have complicated an obvious conce

I appreciate that I've massively over elaborated, but the person I was explaining it to was unsatisfied with the initial explanation that I gave (i.e. the first paragraph of my original post).
Despite that, I'm I correct in what I written?

 

[hackhard]

Despite that, I'm I correct in what I written?

yes

 

[Frank Castle]

yes

Ok, thanks.