# Why is $g$-force pointed downward during upward acceleration?

After a recent physics project in school, where we worked with rollercoasters (it was rather simple questions, about where you would fell the heaviest, lightest etc.) I was reading about g-forces on Wikipedia when I saw this line:

“G-force accelerations ("g-forces" for short) are often expressed as a scalar, with positive g-forces pointing downward (indicating upward acceleration)(...)”

I was reading about it because I wanted to understand how you could feel g-forces as a force downward, when you were really accelerating upwards. Then I found this quote, so I know the facts to my question are correct.

What I’m asking is: Why does a g-force pointing downward (that’s what we feel) indicate upward acceleration? In other words, why do we feel a downward force when we’re accelerating upwards?

G-force is a measurement of force per unit mass that causes the perception of weight. Since it is a measurement of force per unit mass, which is actually acceleration, the term g-force is technically incorrect. It is a measure of acceleration and is equal to 9.8 $$m/s^2$$ at the surface of the earth.

You have the perception of weight simply standing on the ground due to the reaction of the ground to the force of gravity acting on you, even though you are not actually accelerating with respect to the ground. But you are said to be experiencing a g-force of 1 g upwards, even though the force of gravity acts downwards.

Conversely, if you were are in "free fall", as if you were in an elevator whose cables were suddenly cut, you would not have any perception of weight (i.e., you would experience weightlessness) even though you are accelerating with respect to the ground. You are said to be experiencing zero g.

If you stand in on a scale in an elevator at rest, the scale will show your weight here on earth, under the acceleration of 1 g.

Correct. And you are said to be experiencing a g-force of 1 g upward.

But when the elevator is accelerating upward the scale will show a greater weight, and I’m wondering why?

When the elevator accelerates upward the inertia of the person on the scale resists the upward motion so that the elevator floor and scale must push up on the person in order to accelerate the person along with the elevator. The scale has to push upward with extra force to accelerate the person's mass upward.

Applying Newton's second law:

$$F_{net}=ma=-mg+F_N$$

where $$m$$ is the persons mass, $$a$$ the acceleration of person, and $$F_N$$ is the upward force exerted by the scale. Therefore

$$F_{N}=mg+ma$$

Therefore the upward force exerted by the scale is greater than the person's true weight of $$mg$$.

Hope this helps.

• So the acceleration is pointing downwards, because you are being pulled downward by gravity, but the force you feel is a reactionary force produced by the ground? But why would a scale show more weight than in rest if you are accelerating upwards? Sep 12 '19 at 13:08
• Please explain further what you mean by "But why would a scale show more weight than in rest if you are accelerating upwards?" Sep 12 '19 at 13:23
• I finally understand! Thank you. And I would think that when passing over a hill in a rollercoaster you would lift a little from your seat, or even be pressed against the top of you seat (or the “seatbelt” or whatever it’s called). And when you accelerate downwards from the hill you would feel lighter because of the downward acceleration? Anyway, thank you very much :) Sep 12 '19 at 14:17
• @Melvin I think you've got it. Sep 12 '19 at 14:22
• @Sebastiano Thank you very much. Sounds like you went to the same school as me. Ciao! Sep 12 '19 at 22:33

In other words, why do we feel a downward force when we’re accelerating upwards?

You do not directly “feel” the downward force of gravitational attraction due to the Earth which is acting on you.

Considering you as the system then the forces that you “feel” are as a result of other external forces acting on you.
If you are standing in a lift which is accelerating upwards what you “feel” is determined by the upward external force on you due to the lift.
This upward force gives you the “sensation” of you having extra weight ie a downward force acting on you.

In a lift undergoing free fall you will experience the “sensation” of weightlessness as you do not feel any external forces acting on you ie you do not “feel” the effect of the gravitational attraction on you due to the Earth and there are no forces pushing on you due to the lift.

If you are a lift which is accelerating downwards with an acceleration greater than that for free fall there is a downward force on you due to the lift. This force may be from the ceiling of the lift, grab rails in the lift etc.
In this case you have the “sensation” of something trying to push you upwards.

Wikipedia is correct, it is fairly subtle - their are two forces you are mixing up.

(1) The External Force:

The external force is the force being exerted on YOU by the roller-coaster chair beneath you. When you are not accelerating this force is exactly right to cancel the force of gravity trying to accelerate you downwards. When the chair pushes up harder than this you accelerate upwards, when the chair pushes less hard than this you accelerate downwards.

So changes in this force match changes in the direction of acceleration. BUT, this is not the force that people normally report feeling on a roller coaster. How often do people say "we were really accelerating up hard, I felt the keys in my pocket really stab my bum!", instead people will normally talk about a "sinking feeling in their stomach" - these are caused by internal forces.

(2) Internal Forces

To understand internal forces lets go one step "Meta". Instead of thinking about you as a single object lets think about one small part of your body. Lets say you hold your arm out straight in front of you. How much effort is it for you to keep it up against gravity? Not much, but on the roller coaster while you are accelerating upwards it will require more. Just as the chair beneath you needs to exert more force on you to accelerate you upwards you need to exert more force on your arm to accelerate it with you. (If you fail to accelerate your arm at the same rate the chair is accelerating you then your arm will be "left behind": and fall to your side).

So your arm (and all other parts of your body) feel heavier when you are accelerating upwards. This is the "sinking feeling" in the stomach, its your lunch (and organs) feeling heavier as you are boosted upwards. Internal forces result in you feeling heavier - like you are being subject to more gravity. (In a General Relativity sense that is exactly what is happening to you).

We do NOT feel a downward force when accelerating upwards. When accelerating upwards we feel an externally applied upward force (from the seat beneath us). However, what people really notice on roller-coasters is not this external force - what people notice is instead the sense of feeling heavier or lighter. This "heavier/lighter" feeling points the other way.

(In terms of physics terms the "Actual Force" (in an Inertial Reference Frame) on the passenger is upwards. In the passenger's reference frame (which is accelerating upwards relative to the rest frame) their is are "fictitious" forces (called G-forces) that point downward.)

• What about negative g-forces then? (Or the feeling-lighter feeling you get) Sep 12 '19 at 12:35
• I am afraid I do not understand your comment. What about them? It's the same as the feeling heavier example I gave just backwards. (You are accelerating down, therefore if you do not pull your arm down with you it will swing up, therefore your arm feels like its "pulling upwards" - so it feels lighter - which feels like gravity is weaker.
– Dast
Sep 12 '19 at 13:52

It is indeed missleading when written that way (without specifying the frame of reference).

Suppose you are inside a jet-fighter cabin, the g-force is established as the acceleration of your body as viewed from the frame of reference of the cabin (which is at rest in that frame of reference obviously). Your feeling of your body accelerating comes from the feeling you have that the preferential frame of reference is the one of the scenario you are surrounded by; the cabin. But the reality is that the force you are feeling is fictitious. A reference frame with the cabin at rest is a non-inertial reference frame and thus from its perspective other objects (like your body) might seem to experience a force (a force without an apparent source/cause). If you view the same situation from an inertial reference frame (for example, the ground) what you see is the cabin accelerating and your body waiting at rest until it gets pressed by the seat wich is rushing towards your back (no strange forces, only the force of the jet turbines). So, you feel a force that is not there because you generally feel that the correct "world view" is the one of the cabin (because is your surrounding spatial context) and in that "world view" you are accelerating towards the back of the cabin. This acceleration looks like a force but as I said it is fictitious, it comes from the fact that you have inertia and don't move (from the ground reference frame) until the cabin pushes you.

So, the g-force is measured as the acceleration you feel (wich is measured from the non-inertial frame of reference of the cabin) but the true acceleration is the one of the cabin, which indeed moves in the opposite direction (as viewed from the inertial reference frame of the ground). That's why they are opposite (but they have the exact same intensity and timing, the only difference is when you see this from one perspective or the other you see it "reversed").