# What does an acceleration greater than $g$ feel like?

I hope I have put a pretty good title for this question. I was watching Professor Walter Lewin’s lecture on classical mechanics (lecture 7, I think).

Let’s say a person is hanging from a string (by holding onto it) and the string is hanging from the ceiling. That person would feel a tension force $$T$$ which is equal to his weight, let’s say $$mg$$. These are the only two forces acting on the person and since he’s not accelerating, $$T$$ = $$mg$$ in this case. This tension force $$T$$ would give him the perception of his weight, that’s what Professor Lewin said. He said that the tension force $$T$$ by definition is the weight of the person.

Let’s say the person starts accelerating downwards with $$a$$ ($$a$$ $$<$$ $$g$$). In this case $$T$$ = $$m(g-a)$$ and since $$m(g-a)$$ $$<$$ $$mg$$, the person would feel less heavy, as if he has lost weight (that’s what he said). He further said that if that person started accelerating downwards with an acceleration equal to $$g$$, the tension force $$T$$ would be $$zero$$ in that case and he would feel completely weightless. I followed everything up until this point.

But what if the person accelerated at a rate greater than $$g$$? In order for that to happen, we will need another force pulling him downwards. Let’s say there’s a string tied to his legs pulling him downwards with a force $$T_1$$ such that he accelerates at a rate higher than $$g$$, let’s say he is accelerating at $$(g+k)$$ where k is some number. Writing the equation in this case (according to Newton’s 2nd law),

$$T$$ - $$mg$$ - $$T_1$$ = $$-m(g+k)$$

$$=>$$ $$T$$ = $$T_1$$ $$-mk$$

This is what I get. If my math is correct, my question is, what would this tension force $$T$$ feel like for the person? In the first case when the system was not accelerating, the person feels his weight because of $$T$$. When the system was accelerating at the rate $$a$$ $$(, the person felt less heavy. When the system was in free fall, he felt weightless. Now when the system is accelerating at a rate greater than $$g$$, what would it feel like? Would he feel heavier? Or less heavy? I am sure he would not feel weightless. This is my question. Thanks

If you were being accelerated downward at 2g via a rope tied around your legs, you would feel the same as if you were hanging upside down with no acceleration.

When in free fall, you would accelerate downward at g. The rope would be applying a force to your legs to accelerate you at an extra g. This force would be the same as if you were hanging upside down and not moving.

• In the first case, hanging upside down and not accelerating, tension acting on me is $T$ $=$ $mg$. In the second case when I'm accelerating downward at $2g$, $T$ $+$ $mg$ $=$ $2mg$ $=>$ $T$ = $mg$. That's interesting. Can I say that if the rope tied around my legs are accelerating me downward at $4g$, I would be feeling the same way a person with $three$ $times$ my mass hanging upside down and not accelerating would feel?
– 4d_
Nov 23 '19 at 2:20
• @πtimese: Yes. When falling, gravity is accelerating you by $1g$. If you are accelerating downward by $xg$, then the acceleration due to the pull of the rope is $(x-1)g$. This means that the rope is applying a force of $(x-1)mg$. So if $x=4$, then the rope force is $3mg$... equivalent to hanging upside down in a $3g$ environment. Nov 24 '19 at 12:31

Not sure I completely understand the scenario you're describing or maybe some "upwards"s and "downwards"s are swapped. But the key determiner of what things feel like is relative acceleration of different parts of your body. If you're in freefall your arms, legs, kidneys, everything, are all being accelerated together at g so you feel weightless. If you were freefalling towards Jupiter at 318g you would still just feel weightless because everything would be accelerating together at 318g. What we consider normal is the ground pressing on our feet with a force equal to 9.8x our mass. You can take that force away by accelerating the ground downwards (descending very fast on a roller-coaster for example) which puts you back into freefall. Being pulled around by ropes just feels like being pulled around by ropes, and you can't simulate the feeling of weightlessness by attaching ropes to yourself.

• "If you were freefalling towards Jupiter at 318g you would still just feel weightless" Isn't the gravitational acceleration on Jupiter $25$ $\frac{m}{s^2}$ ? What does 318g mean here?
– 4d_
Nov 23 '19 at 2:59
• I just googled the mass of Jupiter and found it was about 318x that of earth. But although you'd be accelerating faster free fall would feel the same as it does here. Nov 23 '19 at 7:45
• For future reference, the surface gravity of Jupiter is indeed 25 m/s$^2$ as stated by @πtimese above. This is because surface gravity is a function of both the mass and the diameter of the planet. Nov 23 '19 at 13:47
• That's interesting and thank you for the correction! Nov 23 '19 at 14:26

Now when the system is accelerating at a rate greater than 𝑔, what would it feel like? Would he feel heavier? Or less heavy? I am sure he would not feel weightless. This is my question. Thanks

It seems to me that the answers to your questions all depend on how one defines the "feeling of weight".

There seems to be general agreement that feeling "weightless" is a feeling associated with the absence of any contact forces on the body, as in the case of free fall. It seems to me that whether or not the presence of any contact force, in any direction on the body, in and of itself is associated with the feeling of weight would depend on just what one means by the "feeling of weight".

The professors's first example of hanging from a string involves a contact force pulling upwards on the body. He said that would give a person a sensation of weight. But it's a different feeling then standing on the ground experiencing a contact force acting upwards on your feet, which I think is more commonly associated with the perception of weight.

Ride a roller coaster and you experience contact forces of various magnitudes and directions on various parts of your body by the structure that restrains you. Do we consider all these contact forces as equivalent to the sensation of weight? If we define the feeling of weight as the feeling associated with any contact force, which the professor seems to be saying with the first example, then I suppose the answer would be yes. As for me, I'm not so sure.

Hope this helps.

• Thanks a lot for the beautiful answer. I agree Professor has defined the feeling of weight as the feeling associated with contact force : Tension in case of hanging from a string, and Normal Reaction in case of standing on the floor of an elevator. But he probably was not implying that he considers all the contact forces that we experience are equivalent to the sensation of weight, as in case of riding a roller coaster. My question is a bit specific, but based on his explanations : What that contact force would feel like for us if we started accelerating downwards at a rate greater than $g$.
– 4d_
Nov 23 '19 at 2:10