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9.3Nonuniform Circular Motion
What about nonuniform circular motion. Although so far we have been
discussing components of vectors along fixed x and y axes, it now becomes
convenient to discuss components of the acceleration vector along the radial
line (in-out) and the tangential line (along the direction of motion). For
nonuniform circular motion, the radial component of the acceleration
obeys the same equation as for uniform circular motion,
a
r
= |v|
2
/r ,
but the acceleration vector also has a tangential component,
a
t
= slope of the graph of |v| versus t .
The latter quantity has a simple interpretation. If you are going around a
curve in your car, and the speedometer needle is moving, the tangential
component of the acceleration vector is simply what you would have
thought the acceleration was if you saw the speedometer and didn’t know
you were going around a curve.
Example: Slow down before a turn, not during it.
Question: When you’re making a turn in your car and you’re
afraid you may skid, isn’t it a good idea to slow down.
Solution: If the turn is an arc of a circle, and you’ve already
completed part of the turn at constant speed without skidding,
then the road and tires are apparently capable of enough static
friction to supply an acceleration of |v|
2
/r. There is no reason why
you would skid out now if you haven’t already. If you get nervous
and brake, however, then you need to have a tangential accel-
eration component in addition to the radial component you were
already able to produce successfully. This would require an
acceleration vector with a greater magnitude, which in turn would
require a larger force. Static friction might not be able to supply
that much force, and you might skid out. As in the previous
example on a similar topic, the safe thing to do is to approach the
turn at a comfortably low speed.
a
r
a
t
a
a
r
a
t
a
a
An object moving in a circle may
speed up (top), keep the magnitude
of its velocity vector constant (middle),
or slow down (bottom).
Section 9.3Nonuniform Circular Motion