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Potential and Kinetic Energy: Roller Coasters, Slides of Acting

Vanguard University of Southern CaliforniaActing

Roller coasters are able to move their passengers very rapidly up and down the hills because the cars gain a large amount of potential energy from the very.

Typology: Slides

2021/2022

Uploaded on 09/27/2022

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Created by LABScI at Stanford
1
Potential and Kinetic Energy: Roller Coasters
Teacher Version
This lab illustrates the type of energy conversions that are experienced on a roller coaster, and as
a method of enhancing the studentsโ€™ understanding of that concept, they will create their own
roller coasters to test out their ideas.
California Science Content Standards:
โ€ข 1. Motion and Forces: Newton's laws predict the motion of most objects.
โ€ข **1l. Students know how to solve problems in circular motion by using the formula for
centripetal acceleration in the following form: a=v2/r.
โ€ข 2. Conservation of Energy and Momentum: The laws of conservation of energy and
momentum provide a way to predict and describe the movement of objects.
โ€ข 2a. Students know how to calculate kinetic energy by using the formula: E = (1/2) mv2.
โ€ข 2b. Students know how to calculate changes in gravitational potential energy near Earthโ€™s
surface by using the formula (change in potential energy) = mgh (h is the change in the
elevation).
โ€ข 2c. Students know how to solve problems involving conservation of energy in simple
systems, such as falling objects.
โ€ข **2h. Students know how to solve problems involving conservation of energy in simple
systems with various sources of potential energy, such as capacitors and springs.
Complete List of Materials (per group):
โ€ข 2 Foam Pipe Insulation Tubes (about 6โ€™ X 7/8โ€™ i.d. x 3/8โ€™ wall) cut in half (see
instructions) which can be found at hardware stores:
โ€ข Masking Tape
โ€ข Marbles
โ€ข Large Wide Space- With available space to tape to.
โ€ข Metric Ruler/Measuring Tape
โ€ข Sharpie Pen
Key Concepts:
โ€ข Energy is the ability of a system or object to perform work. It exists in various forms.
โ€ข Potential energy is the energy an object has inside a force field due to its position. In the
roller coasterโ€™s case, the potential energy comes from its height because the Earthโ€™s force
of gravity is acting on it. Roller coasters are able to move their passengers very rapidly up
and down the hills because the cars gain a large amount of potential energy from the very
first hill.
โ€ข Kinetic energy is mechanical energy that is due to motion of an object.
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Potential and Kinetic Energy: Roller Coasters

Teacher Version

This lab illustrates the type of energy conversions that are experienced on a roller coaster, and as a method of enhancing the studentsโ€™ understanding of that concept, they will create their own roller coasters to test out their ideas.

California Science Content Standards:

  • 1. Motion and Forces: Newton's laws predict the motion of most objects.
  • **1l. Students know how to solve problems in circular motion by using the formula for centripetal acceleration in the following form: a=v2/r.
  • 2. Conservation of Energy and Momentum: The laws of conservation of energy and momentum provide a way to predict and describe the movement of objects.
  • 2a. Students know how to calculate kinetic energy by using the formula: E = (1/2) mv 2 .
  • 2b. Students know how to calculate changes in gravitational potential energy near Earthโ€™s surface by using the formula (change in potential energy) = mgh (h is the change in the elevation).
  • 2c. Students know how to solve problems involving conservation of energy in simple systems, such as falling objects.
  • **2h. Students know how to solve problems involving conservation of energy in simple systems with various sources of potential energy, such as capacitors and springs.

Complete List of Materials (per group):

  • 2 Foam Pipe Insulation Tubes (about 6โ€™ X 7/8โ€™ i.d. x 3/8โ€™ wall) cut in half (see instructions) which can be found at hardware stores:
  • Masking Tape
  • Marbles
  • Large Wide Space- With available space to tape to.
  • Metric Ruler/Measuring Tape
  • Sharpie Pen

Key Concepts:

  • Energy is the ability of a system or object to perform work. It exists in various forms.
  • Potential energy is the energy an object has inside a force field due to its position. In the roller coasterโ€™s case, the potential energy comes from its height because the Earthโ€™s force of gravity is acting on it. Roller coasters are able to move their passengers very rapidly up and down the hills because the cars gain a large amount of potential energy from the very first hill.
  • Kinetic energy is mechanical energy that is due to motion of an object.
  • Thermal energy is energy due to the heat of a system or object. Energy can be converted to heat through frictional dissipation.
  • Friction , or frictional dissipation, is a phenomenon in which mechanically useful energy, such as the motion of the roller coaster, is converted to mechanically useless energy, such as heat or sound. Friction acts on all moving objects, and it is the reason that a ball rolled across an open space will eventually slow down and stop.
  • โ€œConservation of Energyโ€ is a fundamental principle that energy cannot be created or destroyed. Rather, it is transferred between different forms, such as those described above.

Preparation and Lab Notes:

  • Cut 2 pieces of foam insulation along the sides using scissors. (Note: There may be already one side outlined for cutting). Cut the foam insulation evenly in half.
  • In order to minimize the amount of energy loss due to friction in this lab, it is necessary to construct hills that are taut rather than floppy. This is very important to the successful execution of this lab. Taut hills can be made by applying a continuous upward โ€œpullโ€ on the top of each hill. Similarly, the initial downward ramp can avoid โ€œfloppinessโ€ by taping the first part of it flush against the wall. Taping the structure at multiple points to the floor or objects such as chair legs is very helpful.
  • The roller coaster course can be structurally maintained by taping the hills and loops to other objects in the classroom, such as chairs and tables. This will help with accurate height measurements for these obstacles.

Introductory Lecture:

โ€œEnergyโ€ is a term that is ubiquitous in everyday conversation, but it has a specific scientific definition. It is the ability of an object or system to do work. An example might be the kinetic energy that a moving roller coaster car has as it speedily travels on a downhill track. That energy can be used to do the โ€œworkโ€ of carrying the car to the top of a subsequent hill. Similarly, the potential energy the car possesses at the top of that hill allows it to do the work of accelerating to a fast speed on the next downhill. A fundamental principle of the physical world is that energy can neither be created nor destroyed. This principle is known as โ€œconservation of energyโ€. Still, energy can be transferred between different forms, as it is in the previous roller coaster example (kinetic to potential to

Diagram 1

  1. Have one partner form a hill, with its peak located 1 m away horizontally from the starting point as shown in Diagram 2. Do this by pulling up on the insulation to form the peak of the hill. It is helpful to tape the peak of the hill to a chair leg to hold it steady. As one partner holds it still, have the other partner drop the marble from 60 cm vertical height. Diagram 2 (Note: The potential energy of the marble with mass m that starts at height h is equal to mgh. There is no kinetic energy initially if it starts at rest.) QSA1. If there are no energy losses due to friction, what is the maximum hill height you expect it to climb? 60 cm
  2. If the marble makes it over the hill, then raise the height and retry. If it does not roll over, lower the hill and retry. Repeat this process until the maximum hill height is determined (i.e. the marble nearly stops at the top of the hill.) Record it in the table below.
  3. Now drop the marble from 120 cm. Determine the maximum height of the hill and record it in the table below: Dropping Height / cm Hill Height/ cm QS1, QSA2. How does the increase in dropping height affect the maximum hill height? The maximum hill height should increase with increased dropping height. It would double with this increase in the absence of friction, which is not the case. QS2, QSA3. Is your dropping height larger than the height of the hill? Why do you think this is the case? Frictional dissipation accounts for this energy loss. START= 60 CM PREDICTION DIAGRAM: 1 METER

TAPE HERE:40 CM?

PULL ON IT HERE!

Part 2: Energy Dissipation

  1. Using the same maximum hill height as when you dropped the marble from 120 cm, stretch the hill out further from the wall by 50 centimeters (so the center of the hill is 150 centimeters from the wall). This should result in a more gradual slope of the hill up to the same height as in the previous test.
  2. If the marble makes it over the hill, then raise the height and retry. If it does not roll over, lower the hill and retry. Repeat this process until the maximum hill height is determined (i.e. the marble nearly stops at the top of the hill.) Record it below: Maximum hill height at 150 cm separation =____________________ QS3. Is your maximum hill height at 150 cm separation larger or smaller than the maximum hill height at 100 cm separation? - Smaller QS4. Why do you think this is the case? The longer the marble is on the track, the more energy it will lose from the friction. This loss of energy will make the maximum hill height shorter on the longer the track.
  3. Now, tape down the insulation 90 cm away from the starting point as shown in the diagram below, but keep the hill height the same as the maximum height determined in the previous step. (Note: This should give you a steeper slope.) Diagram 3 START= 120 CM FRICTION DIAGRAM: 150 CM TAPE HERE:40 CM TAPE HERE:90 CM FROM THE START POINT
STUDENT ADVANCED VERSION ONLY

The energy dissipated from friction is due to a frictional force , Ff. If an object travels a distance x , then the frictional dissipation of energy is equal to Force (Ff) ร— Distance ( x ) QSA4. When considering frictional dissipation, do you expect the maximum hill height to be larger or smaller than the previous step? The maximum hill height will be smaller.

Created by LABScI at Stanford^7

STUDENT VERSION ONLY
  1. Try dropping the marble through a loop that is small enough for the marble to get through the highest point. If it does not complete the loop, then lower the loop size until it succeeds. _Actual maximum loop height = _____________________ If an object is to continue through a vertical loop, it is not enough for it to merely reach the highest point of the loop. It must actually have a minimum non-zero velocity (it must be moving) along the track at the top point in order to stay in contact with the loop. This velocity depends on the acceleration due to gravity, g , and the radius of the loop, r : v (^) min= gr
  2. Calculate the radius of your loop. Measure how high your loop is and divide by 2: _Radius = _____________
  3. The acceleration due to gravity is 9.8 m/s 2 (980 cm/s 2 ). Calculate the minimum velocity the marble must have to complete the loop: _Predicted minimum velocity = ________________ STUDENT ADVANCED VERSION ONLY
  4. Try dropping the marble through a loop that is small enough for the marble to get through the highest point. If it does not complete the loop, then lower the loop size until it succeeds. _Actual maximum loop height = _________________ If an object is to continue through a vertical loop, it is not sufficient for it to merely reach the highest point of the loop. It must actually have a minimum non-zero velocity along the track at the top point in order to stay in contact with the loop. This is due to the effect of gravitational acceleration that pulls the object vertically downward and off the circular shape of the loop. The minimum velocity is determined by the following force balance: Centripetal Force is the total force acting on an object as it moves in a circle, with acceleration directed toward the center of the circle. If the object has velocity, v , around a circle of radius r , the corresponding centripetal force is given by: โ‚ฌ Fc = mv^2 r

This force must be equal to the sum of external forces acting on the object, which in our case is the gravitational force plus the force that the wall of the slide exerts on the marble, known as the Normal Force. The normal force must be greater than zero (directed downward with gravity), at the top of the loop, if the marble is to stay on the track at that point. This means that the minimum velocity at the top of a loop is that of an object in a circle with a normal force equal to zero ( Fnormal = 0). QSA8. Solve the force balance to write an expression for this minimum velocity in a loop:

v min = gr
  1. The kinetic energy of a rolling marble is related to its velocity, v : 2 2

Ekin = mv At the top of the loop, the potential energy is given by: Ekin = mgh = 2 mgr Conservation of energy requires that the total energy at the top of the loop (kinetic and potential) is equal to the potential energy of the marble at the point of dropping. ๐ผ๐‘›๐‘–๐‘ก๐‘–๐‘Ž๐‘™ ๐ธ๐‘›๐‘’๐‘Ÿ๐‘”๐‘ฆ ๐‘ƒ๐‘œ๐‘–๐‘›๐‘ก ๐‘œ๐‘“ ๐ท๐‘Ÿ๐‘œ๐‘๐‘๐‘–๐‘›๐‘” = ๐น๐‘–๐‘›๐‘Ž๐‘™ ๐ธ๐‘›๐‘’๐‘Ÿ๐‘”๐‘ฆ (๐‘‡๐‘œ๐‘ ๐‘œ๐‘“ ๐ฟ๐‘œ๐‘œ๐‘) QSA9. Write an expression for the maximum allowable height of a loop that a marble Fg + Fnormal Fc ๐น! + ๐น!"#$%& = ๐น! ๐‘š๐‘” + ๐น!"#$%& =

QSA12. If you increased the size of the marble, and therefore its mass, how would the potential and kinetic energy change? Both the kinetic and potential energies would increase. QS9, QSA13. Why is the first hill on all the roller coasters always the highest one? In the absence of additional propelling forces along the track, it is impossible to achieve the initial height if there are any frictional losses, which is always the case in the real world. Part 3: Competition - Build your own rollercoaster! Now, you will have a competition to see who can make the best roller coaster! As you know, the most fun roller coasters are those that send the riders over the highest hills. The group who can make the most number of hills with the highest combined height (add up all the heights - hills need to touch the ground in between) on their rollercoaster will win!