## 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.

      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: a c = v 2 / R F c F F N N = ma c ⇒ F c = mv 2 / R + W = F c (W = mg and F + mg = mv 2 / R c = mv 2 / 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

    In order to calculate the minimum velocity required by the roller coaster car to go through the loop. We assume F N equals to 0. F N + mg = mv 2 / R ⇒ mg = mv 2 / 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

         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= V ≈ 9.81 × 30 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

  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.

 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.

 "Analysis of Situations in Which Mechanical Energy Is Conserved." The Physics Classroom. Web. 21 Apr. 2011. .

 "Energy Transformation on a Roller Coaster." The Physics Classroom. Web. 21 Apr. 2011. .