# Frequently Made Errors: Pseudo and Resultant Forces

### 1. Real versus Fictitious

Pseudo, or “**fictitious**“, forces can arise when a non-inertial frame of reference is used. Using a non-inertial frame makes the usual force/acceleration laws fail. Pseudo forces must be added in to correct them.

### 2. Applied versus Resultant

The applied forces on an object are the specific forces exerted on it by other objects. A resultant force is a force which would be equivalent to some combination of applied forces.

Pseudo forces act as though they are applied forces.

### 3. Inertia

For an accelerating mass *m* in an inertial frame we have the standard equations ##\vec {F_{net}} = \Sigma \vec F = m\vec a##.

If instead we use a reference frame with acceleration ##\vec {a’}##, the apparent acceleration will be ##\vec a – \vec {a’}##. To make the force equation right we have to add in a force ##\vec F_{a’} = -m\vec{a’}## to obtain

##\vec {F_{net}’} = \vec {F_{net}}+\vec F_{a’} = m\vec a-m\vec {a’}= m(\vec a – \vec {a’})##.

In the special case of a frame based on the mass itself, ##\vec a = \vec{a’}##, giving ##\vec {F_{net}’} =0##.

### 4. Centrifugal and Centripetal Forces

These are two ways of describing the force on an object associated with its movement in an arc.

- Centripetal view

*X “In an inertial frame, the centripetal force is the applied force that makes the object move in an arc.”*

Centripetal force is a**resultant**force, not an applied force.✓ “In an inertial frame, real applied forces have a real resultant force producing all the acceleration. The component normal to the velocity is termed the centripetal force.”

*Degenerate example*: A satellite in a circular orbit experiences the applied force of gravitational attraction to its host. Since there are no other applied forces, this equals the resultant force. Since the orbit is circular, this force is always perpendicular to the velocity, so it constitutes the centripetal force.[Arguably, it is better to avoid the term centripetal force altogether and only refer to centripetal acceleration.]

- Centrifugal view

In the reference frame of the circling object, there is no radial acceleration. To get the forces to balance, we need to invent an applied force equal and opposite to the inertial frame’s centripetal force. This is termed the centrifugal force.

Example: *A motorcyclist on a “wall of death” experiences an increased reaction force from the saddle. This provides the centripetal force. To the motorcyclist, it feels like she is being pressed against the saddle by a centrifugal force.*

### 5. Coriolis Force

Example: *A skater spinning with arms outstretched draws his arms in to his chest.*

Viewed in an inertial frame, conservation of angular momentum about the skater’s axis increases his angular speed.

To the skater, a Coriolis force exerts a torque acting on his arms.

**Some prefer to avoid the terms ‘centrifugal force’ and ‘Coriolis force’, referring instead to centrifugal and Coriolis accelerations. It’s a matter of taste. To an individual experiencing a rotation, it does feel like a force. Since it is not an actual applied force, it is called a fictitious force.*

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

The distinction between applied and resultant force may be confusing. And even misleading, when you say that the centripetal force cannot be "applied" but only resultant. What if there is only one force acting on the body in circular motion? What if there is a tangential force as well as a radial force? Will their resultant be the centripetal force?

I never understood all this buzz about "fictitious forces", or a statement like " Using a non-inertial frame makes the usual kinematic laws fail". No, it does not. The equation of motion just doesn't transform covariantly under (linear or rotational) accelerations, as it does under Galilei transformations. That's the whole deal. And because one can state Newton's laws, just as special or General relativity, in a coordinate-free way using differential geometry, there is no reason to single out inertial frames of reference. When there are no forces, a particle follows a geodesic in Newtonian spacetime. This is a straight line, because space is flat. One can assign coordinate values to this line, giving that the second time derivative of x vanishes. If you then apply e.g. a time-dependent rotation R(t), such that x'=Rx, then automatically you get the centrifugal and Coriolis force. No talk about 'fictitious'.

"Centrifugal viewIn the reference frame of the circling object, there is no radial acceleration."Of course there is radial acceleration: v^2/r.

1. Real versus FictitiousPseudo, or “fictitious“, forces can arise when a non-inertial frame of reference is used. Using a non-inertial frame makes the usual kinematic laws fail. Pseudo forces must be added in to correct them.This statement is nonsense by definition of the term kinematics. Kinematics is, by definition, the description of motion without regard to forces. It uses whatever coordinate systems are convenient for the problem at hand, some likely to be inertial while other clearly are not, but all of that is irrelevant.

5. Coriolis ForceA Coriolis force arises from radial motion in a rotating reference frame.This is muddy thinking. There is a Coriolis acceleration that arises at times, but there is no Coriolis force. When the Coriolis acceleration is multiplied by a mass factor, the result is an inertial reaction term, an M*A term, but it is not a force.(And no, Newton's Law does not say that any particular M*A term is a force. It says that the sum of all actual forces is equal to mass*acceleration of the cm for a particle (which is the particle location itself). Randomly chosen M*A terms are not forces, certainly not actual forces.)

A little confused about the applied vs resultant bit too. Having a hard time seeing the gravitational force on an orbitting planet as resultant but not “applied”. Probably an issue with definitions I’m not getting.

Unless its saying that the centripetal is generally resultant. In the case of an orbitting planet, the gravitational force is the only force and it

isan applied force whichresultsin the centripetal force (consisting of only gravitation).The key word in the definition given is that the centripetal force is

aresultant force but nottheresultant force. In general, systems don’t have a single unique resultant force.Newton’s second law says

$$sum_{i=1}^n vec{F}_i = vec{F}_1+vec{F}_2+cdots+vec{F}_n = mvec{a}.$$ The ##vec{F}_i##’s are the forces acting on a body — that is, the applied forces. The centripetal force is not an applied force, so it doesn’t appear as part of the sum of the forces. Note that in an inertial frame, an applied force should have a reaction counterpart, e.g., the Earth exerts a gravitational force on the Moon, and the reaction force is the Moon exerting a force on the Earth. In general, there is no reaction counterpart to the centripetal force because it’s not an applied force.

I agree with the suggestion to avoid the term

centripetal forcealtogether precisely because it leads to the misconception raised here. An object on a curved path instead experiences a centripetal acceleration ##a_c = v^2/r##, where ##v## is the object’s speed and ##r## is the radius of curvature.I was careful (I think) to refer always to the radial component of the resultant.

Exactly.

They don’t? Well, they could reduce to a single force plus a screw … but otherwise?

Interesting suggestion. I’ll add that advice.

In the reference frame of a given object, that object has no acceleration by definition.

The same objection can be made to the term “centrifugal force”. Bear in mind that by definition a fictitious force is one that the observer invents to account for the experience, so it is reasonable to say that there is a Coriolis force. I’ll try to make this clearer.