# Why do pseudo forces not obey Newton's third law?

Why is the statement, that centrifugal force as a pseudo force does not obey Newton's third law (action = reaction) ?

Lets say I'm within a rotation g cage in space, then I feel weight right the same way I would be on Earth. The centrifugal force tends to pull me "down" , but the floor responds with a "reaction" , keeping me at rest with regard to the cage.

Isn't the situation very similar to the situation at earth, where I'm attracted by gravitation instead of a pseudo force? Nobody would identify gravitation as a pseudo force, however, from the equivalence principle bot are the same and cannot be distinguished.

• In your second paragraph, I think the force pair to the normal force is the force your body exerts back on the centrifuge, and not the centrifugal force. Force pairs are found between two different objects. The centrifugal force and normal force on your body and just two forces acting on 1 object. However, I think you found a good way of doing experimental physics. For instance, how would one measure inertial forces (I prefer to use the word 'inertial force' as opposed to pseudo force)? Probably in the way you are thinking about here Commented May 14, 2018 at 12:58
• Then the force which pulls me "down" on the centrifuge (and which gives me a feeling of weight) is not the centrifugal force? What else then? Commented May 14, 2018 at 13:06
• What's tricky here is that the normal force is inherently a reaction force. It's only present when something causes it to be present. But I agree. In the rotating reference frame, there is definitely a centrifugal force pushing down and a normal force pushing up. These forces happen to be equal and opposite. They wouldn't be considered 3rd law force pairs. But then again, they way you are thinking about it (experimentally and practically), I don't think there is a problem with that. It's probably useful to think that way. Actually I probably would. It's easier to consider pseudo forces Commented May 14, 2018 at 13:20
• when there is no normal/tension/contact forces present. For instance, consider an accelerating bus. A block is in the middle of the aisle. As the bus accelerates, the block slides back. Until the block hits the back of the bus, there is nothing stopping the pseudoforce. It's only when it hits the wall does the normal force give an equal/opposite force to the pseudoforce Commented May 14, 2018 at 13:21
• "Why do pseudo forces not obey Newton's third law?" Because it doesn't exist. You can make up any random force that doesn't follow the laws of nature. Commented May 14, 2018 at 13:36

The centrifugal force tends to pull me "down" , but the floor responds with a "reaction" , keeping me at rest with regard to the cage.

A number of ways to test if two forces are not an action-reaction pair:

• Do the two forces act on two different bodies? Here, the body in both cases is question is you: The upward force on you by the floor versus downward force on you be the fictitious centrifugal force. These are not an action / reaction pair.

• Are the forces always equal but opposite? Suppose you try to jump. The normal force will increase, but the centrifugal force remains the same. These are not an action / reaction pair.

• Does the action / reaction pair result from the same interaction? In this case, one of the forces is the normal force while the other is the fictitious centrifugal force. Once again, these are not an action / reaction pair.

There is a reaction to the floor pushing you up via the normal force, and that is you pushing the floor down via the normal force. You can see this in action (not recommended) by washing a load of jeans and T-shirts. After the water drains but before the spin cycle starts, rearrange the contents so the jeans are on one side and the T-shorts on the other. The unbalanced nature of the reaction can make the washing machine bounce around, perhaps even walk.

What about the reaction to the centrifugal force? There is none. The fictitious centrifugal force is the odd force out. It cannot be paired with some other force. More importantly, what centrifugal force? An inertial observer sees only one force acting on you, the normal force exerted by the floor. There is no centrifugal force. It's a fiction, designed so that we can conveniently use Newton's first two laws of motion in a domain where, strictly speaking, those laws does not apply.

Isn't the situation very similar to the situation at earth, where I'm attracted by gravitation instead of a pseudo force? Nobody would identify gravitation as a pseudo force, however, from the equivalence principle bot are the same and cannot be distinguished.

In general relativity, gravitation is indeed a pseudo force. In Newtonian mechanics, it is a real force. The Newtonian reaction to gravity pulling you down is you pulling the Earth up gravitationally.

• So what is correct that gravity has reaction pair (3rd law) or not? I am of the opinion that if there is a way for inertial forces to satisfy 3rd law, that resolves these apparent contradictions (esp. for gravity). Commented Mar 3 at 19:36
• Commented Mar 3 at 21:56

I would like to give an unconventional answer to this.

Newton's 3rd law (in its limited and simplistic statement of only a pair of forces) can be violated in many cases (inertial/apparent forces being one of them).

Specifically:

1. Electromagnetic interactions can violate the 3rd law.
2. Statistical mechanics can violate the 3rd law.
3. Inertial forces violate the 3rd law.
4. Gravity being an inertial force (in relativity) can violate the 3rd law.

The answer to the above is that the 3rd law needs to be expressed in a more general form, and in this form it is found to still hold even in previous cases.

For example:

1. The electromagnetic field carries momentum which when taken into account restores the validity of the 3rd law.
2. Entropic gradients present carry momentum which when taken into account restore the validity of the 3rd law.
3. Gravity although equivalent to inertial force in relativity, is also related to gravitational metric and curvature, which similarly to the EM field can carry and store momentum.
4. Inertial forces can be recast (a-la relativity) as gravitational fields or simply take into account the momentum carried by inertial forces, thus similarly a suitable statement of the 3rd law (in terms of momentum) retains its validity.

All of the above are special cases of conservation of momentum (a suitable generalization of Newton's 3rd law) which by Noether's theorem always holds if the Lagrangian of the (whole) system has certain symmetries (ie translation-invariance).

This conservation [of momentum] law applies to all interactions, including collisions (both elastic and inelastic) and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.

Conservation of Momentum

Even though it is one of the fundamental laws of physics, Newton's third law can be violated in certain nonequilibrium (out-of-balance) situations. When two objects or particles violate the third law, they are said to have nonreciprocal interactions. Violations can occur when the environment becomes involved in the interaction between the two particles in some way, such as when an environment moves with respect to the two particles. (Of course, Newton's law still holds for the complete "particles-plus-environment" system.)

What happens when Newton's third law is broken? - Phys.org

• I think this answer could be improved by making Example 3 and 4 its central idea and making a clearer statement: if a frame measures a pseudoforce, that frame measures an associated field which interacts with the accelerated object to conserve energy and momentum.
– g s
Commented Mar 4 at 6:30
• I'll see if I can find a different phrasing. The answer is inspired by a claim that what tells apart inertial from "real" forces is failure of the 3rd law. Which I think is here adequately addressed. Commented Mar 4 at 6:55