S2.13 Fictitious forces
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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:
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).
Analysis


Analysis
- Consider two reference frames A and B
- A is inertial
- B is accelerating with respect to A
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- 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.
- Then behaviour of the body w.r.t. NI obey’s Newton’s Laws but with a fictitious force
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
is the push you feel the car seat give you.
Check It!
- Are the units OK?
- Does it make sense?
- This is a special case of [A] where
is the centripetal acceleration.
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- 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. | |
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.
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.
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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’
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- Bird crosses center
- It leaves mark
- A has moved round
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- Bird crosses rim again, at B
- It leaves mark at B
- A has moved round further
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The rotating frame perspective:
- Just join up the marks …
- The ‘path’ is curved
- The Coriolis force is ‘invented’ to explain this.
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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:
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Self-Test Questions: | |