### Roller Coaster * Real World Physics Problems

```Centripetal force
and conservation of
kinetic energy and
potential energy
Mengjiao Zhang
Linzi Wang
Silu Gao
Q: What is the most thrilling game in the
amusement parks?
A: The Roller coaster!
It deals with danger and fear; roller coaster
cars go up and down, and go through loops
and twists. The roller Coaster is a highly
exciting entertainment in the amusement
park. However, there are lots of safety
concerns behind roller coaster. Designers of
roller coaster have to apply many physical
concepts to make sure that passengers are
entertained and safe at the same time while
they are on the roller coaster. There are two
basically physics ideas related: centripetal
force and conservation of kinetic energy and
potential energy.
The roller coaster works by gravity. There are no motors used to power it
during the whole ride. Starting from rest, it simply descends down a steep
hill, and converts the (stored) gravitational potential energy into kinetic
energy, by gaining speed. A small amount of energy is lost because of the
friction. That’s the reason why it is impossible for a roller coaster to return
to its original height after the ride is over.
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According to Newton's second Law (F=am), it explains the
reason why circular motion occurs. An object accelerates
toward the center of a circle because of the action of a net
force in that direction
Centripetal acceleration: ac = v2/R
Fc = mac ⇒ Fc= mv2/R
FN + W = Fc (W = mg and Fc= mv2/R)
FN + mg = mv2/R
as the velocity of the roller coaster is greater, the normal
force will be greater so that the roller coaster car will not fall
when it is going through the loops
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In order to calculate the minimum velocity required
by the roller coaster car to go through the loop. We
assume FN equals to 0.
FN + mg = mv2/R ⇒ mg = mv2/R
the mass of the roller coaster car (with passengers
on it) does not play a role in here because the mass
cancels out on both sides
Then the minimum velocity required v =
gR
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Mass of the roller coaster: 700kg
Maximum height of the roller coaster: 80 meters
Radius of the loop of the roller coaster: 30 meters
Gravity of earth is also known to be 9.8ms-2 approximately
In order to find the minimum velocity which the roller coaster
car must have in order to pass the loop, the above values
have to be substituted into the formula v = gR .
Note the minimum velocity can be calculated regardless of
the mass of roller coaster
v=
gR
V= 9.81 × 30
V≈ 17.16 ms-1
At the top, the roller coaster has the largest quantity
of potential energy. Potential energy is the energy of
vertical position which dependent upon the mass of
the object and the height of the object. PE = mass *
g * height
As the cars descend the first drop they lose much of
potential energy due to their loss of height. The
potential energy of this car transfer to the kinetic
energy subsequently. Kinetic energy is the energy of
motion which is dependent upon the mass of the
object and it speed. KE = 0.5 * mass * (speed)^2
The train of coaster cars speeds up as they lose
height. In that way, their original large potential
energy (due to their height) transform into kinetic
energy (due to their speeds). In the whole process of
this ride, cars are losing and gaining height
continuously. In real world, there is also a small
amount of energy lost due to the friction.
The roller coaster illustrate:
KEinitial + PEinitial + Wexternal = KEfinal + PEfinal
(conservation of mechanical energy)
Related notes:
Total energy = ∆KE + ∆PE
= ( 1/2m vf2 - 1/2m vi2)+ ( mghf- mghi)
In order to explain how does these energy transformation clearly,
numerical values will be involved. Assume that the mass of roller
coaster is 700kg. The maximum height of this roller coaster is
80meters. How can we find the maximum speed of this roller coaster
during the whole ride? And where does the maximum speed happen?
By conservation of mechanical energy
The total mechanical energy at the maximum height (the roller coaster
is immobile)
equals its potential energy = mgh = 700*9.8*80 = 548800 j
When the roller coaster is at the lowest position of the whole ride, it
get the maximum speed, because all the potential energy transfers
into kinetic energy.
Kinetic energy = 1/2mv^2= 0.5*700*V^2=350V^2=5488000
solve for V= 35.598m/s
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Energy cannot be eliminated or created, it
could only transform from one form to the
other.
The design of roller coaster contains physics
concepts like gravity, centripetal force,
acceleration, friction and energy conversion.
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Those basic physics elements are protecting
our safety when we are spinning, rising and
falling on that roller coaster.
These data must be carefully collected and
calculated. Physics is a precise science. Roller
coaster is a dangerous facility, one mistake
could cost a huge disaster.
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"Analysis of Situations in Which Mechanical Energy Is Conserved." The Physics
Classroom. Web. 21 Apr. 2011.
<http://www.physicsclassroom.com/class/energy/u5l2bb.cfm>.
"Energy Transformation on a Roller Coaster." The Physics Classroom. Web. 21 Apr. 2011.
<http://www.physicsclassroom.com/mmedia/energy/ce.cfm>.
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