**More circular motion**

10-1-99

Sections 5.3 - 5.5

**Cars on banked turns**

A good example of
uniform circular motion is a car going around a banked turn, such as on a
highway off-ramp. These off-ramps often have the recommended speed posted; even
if there was no friction between your car tires and the road, if you went
around the curve at this "design speed" you would be fine. With no
friction, if you went faster than the design speed you would veer towards the
outside of the curve, and if you went slower than the design speed you would
veer towards the inside of the curve.

In theory, then,
accounting for friction, there is a range of speeds at which you can negotiate
a curve. In most cases the coefficient of friction is sufficiently high, and
the angle of the curve sufficiently small, that going too slowly around the
curve is not an issue. Going too fast is another story, however.

The textbook does a good
analysis of a car on a banked curve without friction, arriving at a connection
between the angle of the curve, the radius, and the speed. The speed is known
as the design speed of the curve (the speed at which you're safest negotiating
the curve) and is given by:

Consider now the role
that friction plays, and think about how to determine the maximum speed at which
you can negotiate the curve without skidding. The first thing to realize is
that the frictional force is static friction. Even though the car is moving,
the car tires are not slipping on the road surface, so the part of tire in
contact with the road is instantaneously at rest with respect to the road.
Also, if we're worried about the maximum speed at which we can go around the
banked turn, if there was no friction the car would tend to slide towards the
outside of the curve, so the friction opposes this tendency and points down the
slope.

The diagram, and a
free-body diagram, of the situation is shown here. Note that the diagram looks
similar to that of a box on an inclined plane. There is a critical difference,
however; for the box on an inclined plane, the coordinate system was parallel
and perpendicular to the slope, because the box was either moving, and/or
accelerating, up or down the slope. In this case the coordinate system is
horizontal and vertical, because the centripetal acceleration points horizontally
in towards the center of the circle and there is no vertical component of the
acceleration.

Moving from the
free-body diagram to the force equations gives:

This can be rearranged
to solve for the normal force:

Note that we're solving
for the maximum speed at which the car can go around the curve, which will
correspond to the static force of friction being a maximum, which is why it's
valid to say that .

In the x-direction, the
force equation is:

Substituting , and plugging in the equation for the
normal force, gives:

There is an m in every
term, so the mass cancels out. The equation can then be rearranged to solve for
the maximum speed:

Note that when the
coefficient of friction is zero (i.e., the road is very slippery), the maximum
speed reduces to the design speed,

and for certain combinations of theta and the
coefficient of friction (both large, in general) the denominator turns out to
be negative, implying that there is no maximum speed; in those cases, you could
drive as fast as you wanted without worrying about skidding. Note that this
does not apply to standard highway off-ramps! The appropriate conditions for no
maximum safe speed (or at least a very high maximum) would be found at
racetracks like the Indianapolis Speedway, for example.

**Vertical circular motion**

Some roller-coasters
have loop-the-loop sections, where you travel in a vertical circle, so that
you're upside-down at the top. To do this without falling off the track, you
have to be traveling at least a particular minimum speed at the top. The
critical factor in determining whether you make it completely around is the
normal force; if the track has to exert a downward normal force at the top of
the track to keep you moving in a circle, you're fine, but if the normal force
drops to zero you're in trouble.

The normal force changes
as you travel around a vertical loop; it changes because your speed changes and
because your weight has a different effect at each part of the circle. To keep
going in a circular path, you must always have a net force equal to mv^{2} /
r pointing towards the center of the circle. If the net force drops below the
required value, you will veer off the circular path away from the center, and
if the net force is more than the required value you will veer off towards the
center.

Consider what happens at
the bottom of the loop, and compare it to what happens at the top. At the
bottom, mg points down and the normal force points up, towards the center of
the circle. The normal force is then not simply equal to the weight, but is
larger because it must also supply the required centripetal acceleration:

This is why you actually
feel heavier at the bottom of a loop like this, because your
apparent weight is equal to the normal force you feel.

At the top of the loop,
on the other hand, the normal force and the weight both point towards the
center of the circle, so the normal force is less than the weight:

If you're going at just
the right speed so mv^{2} / r = mg, the normal force drops to
zero, and you would actually feel weightless for an instant. Faster than this
speed and there is a normal force helping to keep you on the circular path;
slower and the normal force would go up, which means you'll fall out of the
coaster and/or the coaster will fall off the track unless you're strapped in
and the coaster is held down to the track.