# Why am I not accelerated by the reaction force applied by Earth on me? [duplicate]

Newton's third law says that "to every action, there is always an equal but opposite reaction". And Newton's second law of motion says that, $F=ma$ $=>$ $a=\frac{F}{m}$. People says that If I stand on earth's surface, I am applying a force on earth which is equal to my weight and Earth, in return, is applying force to me which is equal in magnitude to my weight but opposite in direction. I know that these forces will not cancel each other out since they are applied on different bodies and not a single but I ask if net force is not zero then according to the equation, $a=\frac{F}{m}$, I must have some acceleration produced in my body. Because the equation is telling us that the force applied on me is same as my force applied to earth but since masses are different and acceleration is inversely proportional to the mass then I must experience some acceleration as my mass is lesser than earth and so acceleration produced must be greater.

• The earth pulls you via gravity (your weight) and it applies an equal but opposite force (against your feet when standing). These two balance out so the net force is in fact zero... – lemon Feb 26 '15 at 13:53
• Well if I take your case then I see that both forces are applied by same body viz., Earth. Since you said that earth is pulling me via gravity and simultaneously you said that earth is applying force against my feet. – user73555 Feb 26 '15 at 13:57
• What am I doing here? Nothing? – user73555 Feb 26 '15 at 13:58
• Imagine you're standing on a tightrope, can you see how the forces balance? Gravity pulls you down which stretches the tightrope and causes the tightrope to exert an equal but opposite force. Right? – lemon Feb 26 '15 at 14:00
• The rope will apply force on what? me or the Earth? – user73555 Feb 26 '15 at 14:04

Why am I not accelerated by the reaction force applied by earth on me?

Because the net force on your centre of mass is zero.

The upward force on your feet is of the same magnitude as the downward force of gravity.

Your major leg bones and spine are in compression because of the opposing forces.

I know that these forces will not cancel each other out since they are applied on different bodies and not a single

You are mistaken. The planet's gravity acts on your body. The reaction force from the ground also acts on your body.

masses are different

Yes your body exerts an attractive gravitational force on the planet and your feet apply an opposite repulsive force to the planet.

However if your reference frame is the planet's local surface and the planetary mass is many orders of magnitude greater than yours, you can ignore your effects on the planet as a whole.

• The downward force of gravitation and the upward normal force are different forces. They are not third law pairs. Unless you are at the North or South Pole, they are not equal in magnitude. – David Hammen Feb 26 '15 at 23:36

In the framework of General Relativity, where the inertial frames are the ones in free fall, you can think that the Earth is accelerating upward, so it is not you who is pushing on Earth but it is Earth that is "running you over" because of its accelerated motion. Luckily enough, if we are standing on ground, we can avoid impulsive forces and our bodies are capable of sustaining a constant acceleration equal to $g$. The fact that we then cannot penetrate Earth as it accelerates upward is because the ground is pushing us up with a force equal to our bodies inertia.

So to make this a bit clearer, flies which are standing on the windscreen of an accelerating car will be squeezed against the windscreen like they have gained extra weight (the impulsive forces are those that would splatter any fly that wasn't already on the windscreen and that happens to smash against it because the car is moving towards them). Here we're focussing on the extra weight from acceleration alone, not on wind resistance.

• I didn't find your answer easy to understand. I am a student of high school. Can you tell me the way so I can understand it. – user73555 Feb 26 '15 at 14:34
• In a nutshell you should think that the Earth is accelerating upward and you are caught up in this accelerated motion since you stand in Earth's way perhaps this video can help a bit youtube.com/watch?v=E43-CfukEgs – Phoenix87 Feb 26 '15 at 14:36
• I don't think starting the answer with reference to general relativity is very helpful here. – JiK Feb 26 '15 at 14:36
• yes of course! it is. – user73555 Feb 26 '15 at 14:49
• @Phoenix87 gave the more fundamental answer which relates gravitational effects to GR. There's nothing wrong with it. It's simply not what high school physics classes give exposure to. Some HS students will appreciate it, and they will become physicists :). – Bill N Feb 26 '15 at 17:58

I think it helps if you look at the problem from more of an intuitive perspective.

When you're standing still on Earth, you're clearly not accelerating, right? (In this example we are ignoring the energy and force exerted by the molecules in our body, which are all moving to some extent regardless of where we are.)

If you were accelerating, you'd see some change in velocity over time, which you clearly don't when you're still. We can say, then, that the net forces of you and gravity are pretty much zero. They are no longer zero, for example, when you jump, or run, etc.

However, you have to keep in mind that this is all from your own perspective (I think that's why one can bring up general relativity as a good talking point here) and the perspective of anyone on Earth looking at you. There are some theories that the Earth is moving really fast in a straight line (like a car, or bullet, or some other projectile) and humans are just "in the way" of that movement, and thus we feel the effect of this thing we call gravity (similar to a person inside an elevator going up; when you're in an elevator, you go up because you're accelerating with the elevator). If this is the case, then both humans and Earth are accelerating relative to everything else, but humans at a stand still are not accelerating relative to the Earth. This also confirms the notion that acceleration is always relative between two independent objects.

• Vague! Really vague... – user73555 Feb 26 '15 at 15:28

I know that these forces will not cancel each other out since they are applied on different bodies and not a single ...

That's correct.

... but I ask if net force is not zero then according to the equation, a=Fm, I must have some acceleration produced in my body.

First off, it's $F=ma$ (or equivalently, $a=F/m$), not $a=Fm$.

One thing you are missing is that Newton's second and third law address different concepts. Newton's second law talks about how the acceleration of object is related to the net force exerted on that object by other objects. Newton's third law talks about the individual force (not net force) exerted by one object on another is related to the different objects exert on one another.

In particular, Newton's second law says $\boldsymbol F_\text{net} = m\,\boldsymbol a$ while Newton's third law says $\boldsymbol F_{a,b;i} = - \boldsymbol F_{b,a;i}$, where $\boldsymbol F_{a,b;i}$ is the force exerted on body $a$ by body $b$ via interaction $i$. The $\boldsymbol F_\text{net}$ in Newton's second law is the sum of all of the individual forces acting on the body.

Another thing you are missing is that other forces act on you other than gravitation when you are standing on the ground. Suppose you stand on your chair and jump off. For a brief period of time between the jump and the landing, you are accelerating. The difference is that you have removed the upward force exerted on you by the ground. This is a different force than gravitation. It is the "normal force". It's what keeps you from sinking into the floor.

To go from Newton's third law to Newton's second law, you have to consider all of the forces acting on an object. This is why students are taught to draw free body diagrams. Forces are vector quantities. That means they add, in the vector sense of addition.

While Newton didn't use vectors (he couldn't have because the concept of a "vector" postdates Newton by 200 years), he did make the distinction between individual forces and net forces in his corollaries to his three laws of motion. Some physics even instructors go to the extent of teaching the vectorial nature of forces as Newton's fourth law.

Why am I not accelerated by the reaction force applied by Earth on me?

Truth be told, you are accelerated by the reaction forces applied by the Earth on you. The Earth is rotating, one revolution per day. Unless you are at the North Pole or South Pole, you are undergoing uniform circular motion about the Earth's rotation axis. The downward force gravitational force the Earth as a whole exerts on you and the upward normal force the Earth's surface exerts on you do not quite cancel. There's a tiny residual non-zero net force that is just enough to make you rotate about the Earth's rotation axis at one revolution per day.

To you, lets apply the second law. As it states, actually
$\sum F=m a$
NOTE THE SUMMATION, which means the resultant force applied. If you're just sitting or not moving, both normal force and gravity exerted in opposite direction, for its magnitude, they equals.
You're still having problem with how will the Earth move ha?
The third law tells, You give the earth some very small force (equal to your weight though), and as the earth exerts normal force on you, you're pushing the Earth also with the equal force (firstly equal to the normal force, and normal force equal to your weight). So If there's only you on the earth, the gravity from you and the normal force you exert on earth is in opposite direction and same magnitude, the Earth don't move either.