If you flip through the pages of just about
any physics textbook, you will probably find
a picture of a roller coaster loop illustrating
the concept of centripetal acceleration.
Textbook problems often portray loops that
are perfectly circular in order to simplify
calculations, but in reality, roller coaster
loops are not circular at all.
The fundamental problem with circular loops
is the constant radius, which generates intense
g-forces that make them uncomfortable and
dangerous for riders.
This was first realized in the 1840’s when
the first centrifugal railways were constructed
with circular loops in Western Europe, and
it wasn’t until alternative shapes were
introduced in the 1970’s that roller coaster
loops started to become popularized.
If you look around at different roller coasters
today, you will see that there are a variety
of loop shapes used by different design companies,
and this video is going to explain the physics
that roller coaster engineers use to design
these shapes.
Before we can do that, though, we need to
start with a sound understanding of the physics
behind simple circular loops.
Let’s use this simple roller coaster as
an example, which consists of a lift hill
and a single loop.
As a train is lifted to the top of the coaster,
it gains gravitational potential energy according
to the equation Eg = m*g*h, where m is the
mass of the train, g is acceleration due to
gravity, and h is the height above a chosen
reference point.
In this scenario, we will choose the bottom
of the vertical loop as our reference point,
and we will denote the height of the lift
hill as ho.
As the train heads down the drop towards the
loop, its potential energy is converted to
kinetic energy which causes it to accelerate.
At any given point, the gravitational energy
possessed by the train can be calculated as
m*g*h, and the kinetic energy can be calculated
as m*g*ho – m*g*h, or simply m*g*(ho – h).
Kinetic energy and velocity are related according
to the equation Ek = ½*m*v2, and setting
our two equations for kinetic energy equal
to each other allows us to solve for the train’s
speed as a function of its height.
By cancelling out mass on each side and rearranging,
we arrive at the expression v = sqrt(2*g*(ho
– h)).
Setting h to zero in this equation gives us
the velocity of the train just before it enters
the loop, and we will denote this as vo.
This equation does not account for the energy
losses due to friction and air resistance
that would cause the train to lose speed as
it travels along the track, but we are going
to neglect energy losses for the time being.
As the train enters the vertical loop, it
begins to experience centripetal acceleration
which acts towards the center of the circle.
The magnitude of this acceleration is equal
to the train’s velocity squared divided
by the radius of the loop, and it is a direct
result of the net force acting on the train.
If we neglect the length of the train and
draw a free-body diagram for an arbitrary
position on the loop, then we can see that
the two external forces acting on the train
are gravity, Fg, and the normal force from
the track, FN.
FN acts at an angle, θ, which is also equal
to the angle of the track at that point.
Using Newton’s second law, which states
that net force is equal to mass times acceleration,
we can consider the sum of forces in the direction
of the normal force to arrive at the expression
FN – Fg*cos(θ) = m*(v2)/r.
We can replace Fg with mass times acceleration
due to gravity, and re-arranging to solve
for the normal force gives FN = m*(v2)/r +
m*g*cos(θ).
Dividing FN by mass and acceleration due to
gravity yields the g-force that passengers
will experience as the train travels around
the loop, and we will denote this as G.
We can now plug in our equation for the train’s
velocity that we derived earlier, and cancelling
out small g leaves us with a simple expression
for g-force that is a function of the train’s
height, h, and the angle of the track, θ.
ho and r are constants in this equation, and
they do not change regardless of where the
train is on the loop.
It is more convenient to express g-force as
a function of a single variable, and we can
do this by relating the height of the loop
at any given point to the angle of the track
at that point.
Since our loop is a perfect circle, we can
use a little bit of trigonometry to arrive
at the expression h = r*(1 – cos(θ)).
For a circular loop, the angle of the track
at any given point is equal to the angle that
the train has traversed about the center,
which makes the math relatively straight forward.
For a non-circular loop, it becomes much more
difficult to relate height and track angle.
Plugging this equation for h back into our
equation for g-force and simplifying, we get
G = 2*ho/r + 3*cos(θ) – 2.
This expression can be used to calculate the
g-force that passengers will experience at
every point around the loop as long as we
know ho and r.
For convenience, we can also replace ho with
vo using the expression that we derived earlier,
so that we can use the speed of the train
just before entering the loop instead of the
initial height.
This is particularly important if we want
to include energy losses because the train
will lose a certain amount of energy before
reaching the loop, and we can adjust vo in
order to account for these losses.
If we wish to plot g-force as a continuous
function along the path of the loop, then
we can also replace θ with s/r, which comes
from the arc length equation for a circle.
In this equation, s represents the arc length,
or length of track, that the train has traversed
around the loop.
This final equation can now be used to plot
g-force for any circular loop with radius,
r, when the train has an initial velocity,
vo.
The g-force plot for any loop where the train
has just enough speed to make it all the way
around looks like this, and we can quickly
see the fundamental issue with circular loops.
Regardless of the radius of the circle and
the initial velocity, there will always be
a 6g difference between the bottom and the
top.
If the train has just enough energy to make
it all the way around, then the speed of the
train will be exactly zero at the top of the
loop, and the passengers will experience -1g.
At the bottom of the loop, the passengers
will experience +5g, and this value will be
even higher if the train is travelling faster.
If the train travels at exactly the right
speed for the passengers to experience weightlessness,
or 0g at the top, then they will be experience
+6g at the bottom.
The F24 Committee on Amusement Rides and Devices
has created ASTM standards for the design
of safe amusement attractions, and they provide
g-force exposure limits that include magnitude,
direction, and duration.
For the positive Gz g-force that you would
experience on a vertical loop, the F24 g-force
limits look like this.
If passengers are going to be exposed to 4g,
then the duration of that exposure should
not exceed 4 seconds, and this duration decreases
further as higher g-forces are reached.
This is one of the reasons why the maximum
g-force on a roller coaster does not typically
exceed 4g.
In addition, acceleration in excess of 4g
may also cause some passengers to experience
discomfort.
On a circular loop, the passengers are not
only exposed to very high levels of g-force,
but the magnitude of the g-force also changes
rapidly.
If the train enters the loop from a horizontal
section of track, then the g-force will change
almost instantaneously from 1g to more than
5g because there is no transition between
the infinite radius of the straight track
and the radius of the loop.
The g-force will then decrease by 6g as the
train approaches the top of the loop, and
it will increase by 6g on the second half
before suddenly dropping back to 1g as the
train exits onto another straight section
of track.
For these reasons, it is physically impossible
to design a perfectly circular loop that is
both thrilling and comfortable for riders,
and this is why they are not used on modern
roller coasters.
Instead, roller coaster engineers decide what
g-forces they want passengers to experience
on a vertical loop, and they reverse-engineer
a shape that will provide these forces.
For example, let’s suppose we want to design
a loop that will provide constant centripetal
acceleration all the way around.
The equation for centripetal acceleration
is v2/r, and we will set this equal to a constant
value, C*g, where C is the magnitude of the
desired acceleration in g’s, and g is acceleration
due to gravity.
Rearranging for r, we get r = v2/(C*g), and
since v2 is inversely proportional to height,
this tells us that the radius of the loop
needs to decrease linearly with elevation.
Plugging in our equation for velocity and
simplifying, we arrive at the expression r
= (1/C)*(vo2/g – 2*h).
We will replace h with y at this point in
order to match the x-y coordinate system that
we will use to plot the loop, however y will
still represent height above our reference
point.
Since the radius is now a function rather
than a constant, it becomes more difficult
to define the path of the loop, and we will
need to use a little bit of calculus to do
this.
Let’s start by considering an extremely
short segment of curved track with arc length
Δs that is oriented at an angle θ.
The length of this segment is so small that
the radius of curvature is essentially constant
over its length, which means that we can treat
it as a circular arc with internal angle Δθ.
Using the arc length equation for a circle,
we can write Δs = r*Δθ, or Δθ/Δs = 1/r.
We will denote the horizontal length of the
segment as Δx and the vertical height of
the segment as Δy, and since Δs has negligible
length, we will approximate it with a straight
line to complete a right-angle triangle.
Using trigonometry, we can now write cos(θ)
= Δx/ Δs and sin(θ) = Δy/ Δs.
If we now take a limit as Δs goes to zero,
then we obtain a set of three differential
equations that we can use to define the path
of our loop.
Since we are trying to obtain a loop with
constant centripetal acceleration, we will
plug our previous expression for r into the
first differential equation to obtain ∂θ/∂s
= C/(vo2/g – 2*y).
We can now use this set of equations to create
our loop by solving the system numerically
with computer software.
To plot the shape, we start at the origin
of our coordinate system, and we use discretized
versions of our differential equations to
traverse the loop incrementally in small steps.
I chose to use Euler’s method for this particular
problem, and the set of discretized equations
looks like this.
These equations take the x-coordinate, y-coordinate,
and track angle at our starting position,
n, and they use this information to predict
the coordinates and track angle at the next
position, n + 1.
We can specify the distance that we want to
use between these points, and the smaller
the distance is, the closer our numerical
solution will be to the true analytical solution.
Decreasing the distance between points also
increases the computation time that is required,
however, since more calculations are required
to get all the way around the loop.
Once we have found the coordinates and angle
at the n + 1 position, this then becomes the
new starting point for the next set of calculations.
This process is continued until we have made
it all the way around the loop, leaving us
with a final shape that resembles an upside-down
teardrop.
This particular loop was designed to give
a constant centripetal acceleration of 3g’s
when the train has an initial speed of 20
m/s.
Now that we have plotted our shape, we can
generate a g-force plot along the path like
we did for the circular loop earlier.
For any loop with constant centripetal acceleration,
the g-force at the bottom will always be equal
to C + 1, and the g-force at the top will
always be C – 1.
This means that there will always be a difference
of 2g’s between the top and bottom of the
loop, and this range can be shifted up or
down by changing the magnitude of C.
These g-forces make the constant acceleration
loop much more desirable than a circular loop,
and we can find many examples of actual roller
coasters that use similar shapes.
One example is Carolina Cyclone at Carowinds,
which was built by Arrow Dynamics in 1980.
If we adjust the initial velocity and centripetal
acceleration in our set of differential equations,
then we can plot a shape that almost perfectly
aligns with the two loops in this photo.
The double loops on this roller coaster also
provide some good insight into energy losses,
since the second one is positioned closer
to the ground than the first.
The train will have less energy as is travels
around the second loop due to friction and
air resistance, but decreasing the height
causes more gravitational energy to be converted
to kinetic energy.
This allows the same loop shape to be used
twice since the initial velocity of the train
will be the same at the start of each one.
If the second loop were positioned at the
same elevation as the first, then the shape
would need to be altered by tightening the
radius in order to account for the decreased
speed.
Since trains lose energy gradually as they
travel along the track, in theory the radius
of each loop should decrease proportionally
from the start to the end in order to maintain
constant centripetal acceleration, however
this is often neglected because the energy
loss around a single loop is relatively small.
In addition to loops with constant centripetal
acceleration, roller coaster engineers may
also choose to design loops that provide constant
g-force.
The equation for g-force that we derived earlier
is G = (v2)/r + g*cos(θ), and we can re-arrange
this equation to get a function for r just
like we did before.
Plugging in our expression for velocity, we
get r = (vo2 – 2*g*y)/(G – g*cos(θ)).
We can now plug this equation directly into
the same set of differential equations that
we used before, and we can use Euler’s method
again to solve the system and plot the loop
shape.
This particular loop was designed to give
a constant g-force of 3.5g’s when the train
has an initial speed of 20 m/s.
Although the shape looks similar to our previous
loop, there are actually subtle differences
in curvature that eliminate any variations
in g-force.
By adjusting G in our differential equations,
we can create a shape that will provide any
constant g-force magnitude that we want.
Constant g-force and constant centripetal
acceleration loops are both good alternatives
to circular loops that generate lower levels
of g-force, however we still have not addressed
the sudden change in g-force at the beginning
and end of the loop.
To solve this problem, we need to provide
smooth curves where the radius is directly
proportional to the length travelled along
the track.
This type of curve is known as a clothoid,
or Euler spiral, and it was first implemented
in roller coaster designs by Werner Stengel
starting in 1975.
Clothoids are commonly used to connect roller
coaster elements together that have different
curvatures, and they can also be used to create
vertical loops.
A clothoid is defined using a parametric equation
where x and y are defined as functions of
arc length, s.
These functions contain integrals that cannot
be solved analytically, and we need to use
computer software to solve for x and y using
a numerical integration method.
Carrying out this integration leaves us with
a spiral shape where the radius of curvature
starts at infinity and decreases linearly
along the length of the spiral.
If we take the first segment of the spiral
up to the point where the tangent line first
becomes horizontal, then we can mirror this
segment about that point to create a complete
clothoid loop.
We can create a g-force plot for this loop
like we have done for the previous shapes,
and we can adjust the g-forces by scaling
the loop up or down.
Regardless of the train speed or scale factor,
the g-force at the beginning and end of the
loop will always be equal to exactly 1g to
match the horizontal track on either side.
If a roller coaster engineer wants to design
a loop using clothoid curves, but they do
not like the g-forces generated by a full
clothoid loop, then they may choose to mix
and match segments from different loop shapes.
For example, we can take the bottom portion
of a clothoid loop and add on the top portion
of a constant g-force loop, which will eliminate
the drop in g-force that would be present
for a full clothoid.
There are endless shapes that can be created
in this way, but we do need to make sure that
the curvature of the different segments is
exactly the same at the connecting points,
otherwise the g-force will be discontinuous
between segments.
By combining different curves together, we
can even turn a circular loop into a good
design by replacing the bottom portion with
two clothoid curves.
This design is similar to loops used on actual
roller coasters like Blue Fire at Europa-Park,
which was built by Mack Rides in 2009.
To briefly summarize everything up to this
point, we have taken a look at 3 non-circular
loop shapes, and we have developed a good
method to reverse-engineer a loop to provide
just about any g-force experience that we
might want for our riders.
However, the original derivation neglected
the length of the roller coaster train, and
we still need to discuss how this will affect
our design.
All of the g-force plots that we have looked
at so far are valid, but only for passengers
sitting directly in the middle of the train.
Passengers sitting closer to the front or
back will experience different g-forces because
they will be on different parts of the loop
at any given time.
As a result, each row of passengers will travel
over a fixed point on the loop at a different
speed, and we need to account for this by
shifting the relationship between speed and
radius in our g-force function.
The resulting g-force plot for a loop with
constant centripetal acceleration looks like
this, and we can see that passengers sitting
at the front of the train will experience
higher g-forces on the first half of the loop
and lower g-forces on the second half because
the function is skewed to the right.
The opposite is true for passengers sitting
at the back of the train because the function
is skewed to the left.
In either case, sitting closer to the front
or back will result in higher levels of g-force
compared to the middle of the train.
This is an important consideration when designing
a loop because all of the g-forces need to
be within the allowable limits for safety.
Although the underlying physics of roller
coaster loops is relatively simple, there
is a great amount of mathematics and engineering
involved in creating a design that is both
safe and thrilling for riders.
Each roller coaster design company has come
up with their own formula which makes every
roller coaster unique and exciting, and they
have all come a very long way from the first
circular loops that were built nearly 200
years ago.
Hey everyone, I hope you enjoyed today’s video about the physics of roller coaster loops.
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