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 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 that 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 into 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’, preferring 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.