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S2.13 Fictitious forces

Preamble

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Preamble

  • Newton’s laws describe motion from perspectives of an inertial reference frame.
  • Two questions:
    • What makes a reference frame ‘non-inertial’?
    • How are Newton’s law’s modified in non-inertial reference frames?
  • General answers are expressed in the following claim:
 

Key Point 2.12

Any reference frame accelerating w.r.t. an inertial frame is non-inertial.

When the motion of some body is described from the perspectives of a non-inertial reference frame Newton’s laws hold only if we introduce fictitious forces which reflect the acceleration.

Such forces are indistinguishable from gravitational forces (The Principle of Equivalence).

XThe Principle of Equivalence

Analysis

Analysis

  • Consider two reference frames A and B
    • A is inertial
    • B is accelerating with respect to A
    Graphic - No title - accframes
  • Consider behaviour of body C, free of external forces.
  • In frame A we can invoke Newton laws: C has zero acceleration w.r.t. A
  • But B accelerates w.r.t. A so C will have a non-zero acceleration wrt B.
  • Thus Newton’s laws do not hold for B: so B is non-inertial.
  • We can make Newton’s laws work in B by inventing a force that accounts for the acceleration seen in this frame.
  • This is the origin of fictitious (pseudo) forces.
 

[A] Body at rest in accelerating frame

  • Consider body of mass m stationary in a non-inertial (NI) frame
  • Suppose NI frame has acceleration $\vec{a}_{NI}$ with respect to inertial frame, I.

    Note!

    Note!

    You need to think about the notation here. Recognise that $\vec{a}_{NI}$ is the acceleration of the non-inertial reference frame w.r.t. the inertial frame. Since the body is at rest in NI, it is also the acceleration of the body w.r.t the inertial frame.
     

Graphic - No title - niforcesgen

Analysis

Analysis

Help before you start?

Help before you start?

We have writen the following argument so as to bring out its generality. The price is that it may feel a bit abstract. To make it seem less so you can have the following specific example in mind:
  • The ‘body’ can be yours, and sitting in your car.
  • The car is accelerating along a straight road; it (the car) plays the role of the non-inertial reference frame (NI).
  • The road is the inertial reference frame (I).
  • The force $\vec{F_A}$ is the push you feel the car seat give you.
 
  • Inertial frame perspective

    • acceleration of body: $\vec{a}_{NI}$
    • force on body: $\vec{F}_A$
    • Newton’s 2nd Law:
      \[ \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{m\vec{a}_{NI} =\vec{F}_A} \]
    Graphic - No title - niforcesgenone
  • Non-inertial frame perspective

    • acceleration of body = zero
    • To make Newton’s 2nd Law work we demand:
      net force = zero
    • So we must invent a force $\vec{F}_{NI}$ such that
      \[ \vec{F}_{NI}+ \vec{F}_{A} =0 \]
    • Hence
      \[ \vec{F}_{NI}=- \vec{F}_{A} = -m\vec{a}_{NI} \]
    Graphic - No title - niforcesgentwo

    Help?

    Help?

    Appealing to the example of you-in-your-car, again, $\vec{F}_{NI}$ is the force you feel pushing you back into your seat.
     
Check It!
  • Are the units OK?
  • Does it make sense?
 

[B] Body at rest in a rotating frame

  • This is a special case of [A] where $\vec{a}_{NI}$ is the centripetal acceleration.
Graphic - No title - niforcescentgen

  • Observers in rotating frame must invent a fictitious force which
    • has magnitude
      FNI=mv2/r
    • acts out from centre of rotation
This is the centrifugal force.
Graphic - No title - niforcescent

Warning

Warning

Distinguish carefully between ‘centripetal’ and ‘centrifugal’.

The centripetal force is the ‘real’ force (friction, gravity, tension …) responsible for the acceleration exhibited by a body moving in a circle: it is the force that ‘keeps the body on its circular path’. It acts towards the centre (the meaning of the word ‘centripetal’).

The centrifugal force is a fictitious force which we invent to explain our experience in a rotating frame–namely that, despite being subject to the centripetal force (it is ‘there’ for us in this frame too!), we do not find ourselves accelerating. We invent this force to be equal and opposite to the centripetal force. It acts outwards from the centre.

 
IA bit of Bond...
AA Demonstration of the Centrifugal Force

[C] Body moving in a rotating frame

In addition to the centrifugal force a body moving in a rotating frame ‘experiences’ a second fictitious force which:

  • depends on its speed in the rotating frame;
  • acts at right angles to its path;
  • makes the path curved with respect to the rotating frame.
Graphic - No title - niforcescor

This is the Coriolis force.

Analysis

Analysis

  • Consider incontinent bird moving across rotating bird table.
  • Suppose bird moves at uniform velocity in earth (inertial) frame, and crosses centre of table.

The inertial frame perspective:

  • Bird flies straight.
  • Disc rotates underneath.
  • Consider three points on the flightpath.
  • Bird crosses rim at A.
  • It leaves ‘mark’
Graphic - No title - coriolisarg1
  • Bird crosses center
  • It leaves mark
  • A has moved round
Graphic - No title - coriolisarg2
  • Bird crosses rim again, at B
  • It leaves mark at B
  • A has moved round further
Graphic - No title - coriolisarg3

The rotating frame perspective:

 
  • Just join up the marks …
  • The ‘path’ is curved
  • The Coriolis force is ‘invented’ to explain this.
Graphic - No title - coriolisarg4
 
IThe Coriolis force and wind patterns
AA Demonstration of the Coriolis Force

Learning Resources

Textbook: Conspicuously absent from HRW. Those with a desire to find out more may want to consult Serway and Jewett (Physics, 6th ed) on pages 159-161
Course Questions:
Self-Test Questions: