Rollercoaster Lab.docx 11

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Feb 20, 2024

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Rollercoaster! Measuring Energy and Force on a Rollercoaster PURPOSE In this activity you will measure the potential energy at the top of a ramp, the kinetic energy at the bottom of the ramp, and the energy lost on the ramp. You will then use this value of kinetic energy to determine the height from which the car must roll in order to complete a loop and allow your car to successfully reach the end of the track. MATERIALS Each lab group will need the following : calculator, TI® graphing tape, masking clamp ruler, clear metric meter stick paper, carbon racetrack, extension set racetrack, sets PROCEDURE PART I: TRACK CHARACTERISTICS In Part I of the lab, you will measure the speed of a car as it rolls off the end of a curved track. 1. Set up the apparatus as shown in Figure 1 below. Line up the end of the molding track with the edge of the tabletop. If there is extra molding above the clamp at the top of the ring stand, have one of your lab partners hold it up, or find a way to prop it up so that it does not interfere with Part I of the lab. 1

2. Measure the vertical distance d y from the floor to the end of the track, and the height h at which you released the car above the bottom of the track and record your values in the data table for Part I on your student answer page. 3. Use a tape measure to measure the length of the track L in centimeters from the top of the track at height h to the end of the track (from point A to point B) and record your value in the data table for Part I on your student answer page. 4. Roll the car down the track and allow it to roll off the end of the track and land on the floor. Place a white sheet of paper on the floor at the location where the car landed. Place a piece of carbon paper on top of the white paper with the carbon side down. The next time you roll the car down the ramp and allow it to land on the floor, the car will make a mark on the white paper. You may want to tape the white paper to the floor. Depending on the way the car hits, you may have to make a decision about the actual point of contact. If it lands on the wheels and make 4 marks, use the center. If it hits on the front end use this mark. Other possibilities may require you to decide where the center of mass of the car hits. 5. Measure the horizontal distance dx from the point at which the hanging weight touches the floor to the point at which the car landed on the floor and record your value in the data table for Part I on your student answer page. 6. Repeat step 4 two more times, record your values in the data table for Part I on your student answer page. Use these values to find the average horizontal distance traveled by the car. 7. Answer the Analysis questions for Part I before moving on to the procedure for Part II. PART II: LOOP HEIGHT In Part I of the lab, you determined the speed and kinetic energy of the car as it came off the end of the track, the initial potential energy of the car at the top of the track, the energy lost by the car on the track, and the energy lost per centimeter of track. In Part II, you will predict the height from which the car must be let go on the track in order to make it just go completely around a loop of specific height. You will also predict the speed of the car at the end of the track and confirm this speed by predicting and measuring the speed of the car as it goes off the table and strikes the floor. 1. Clamp one end of the track to the top of the ring stand or other structure used to elevate the ramp. You may use the same set up and same height h as in Part I. If after making your calculations, you may need to increase the height, or you may need to start the car at some distance down the track. The exact distance h from which you release the car will be calculated after other measurements are made. 2. Add the circular loop onto the end of the track and adjust your apparatus so that the end of the track on the outside of the loop ends at the exact edge of the table. 2

3. Use a tape measure or meter stick to measure the same length of the track L in centimeters as in Part I from the top of the track at height h to the beginning of the loop. Measure the diameter of the circle (H) and calculate the circumference of the circle. Also measure any distance from the outbound side of the loop. 4. In the Analysis section for Part II of your student answer page, predict the minimum height h from which the car can be released in order to make it completely around the loop without ever losing contact with the track. Also predict the velocity of the car when it leaves the end or the track based on the amount of kinetic energy the car still posses as it reaches the edge of the track/table and becomes projectile. 5. Roll the car down the ramp and test the accuracy of your prediction. Do not wind the spring in the car, let the car roll freely using only its gravitational potential energy. 6. If the car does not make it around the loop without losing contact with the track, or if you think the car could be moving slower, adjust the height h from which the car is released and roll the car again. 3

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Rollercoaster! Measuring Energy and Force on a Rollercoaster DATA AND OBSERVATIONS PART I: TRACK CHARACTERISTICS Mass m of your car = __0.039__ kg Length L of the track from the top of the ramp to the end of the track = _____154 ____ cm Trial # Initial Height of Car above Table h (m) Distance from Floor to Bottom of Ramp d y (m) Horizontal Distance d x (m) 1 0.51 0.76 0.93 2 0.51 0.76 0.985 3 0.51 0.76 0.92 ANALYSIS PART I: TRACK CHARACTERISTICS In Part I of the lab, we want to find the speed of the car as it leaves the end of the ramp so that we can then find the kinetic energy at the end of the ramp. As the car leaves the end of the ramp, it is moving horizontally and vertically at the same time. In the horizontal direction, it does not accelerate, but moves with a constant speed. In the vertical direction, the car accelerates at 9.80 m/s 2 downward as it falls. We can combine the horizontal and vertical motions of the car to find its speed, v, as it leaves the end of the track. 4

1. The time the car is in the air after it leaves the end of the ramp can be found using the motion equation d y = 1 2 gt 2 where d y is the vertical distance from the floor to the end of the ramp, and g is the acceleration due to gravity. Rearrange this equation for the time of flight t in terms of d y and g. Show your steps in the space below. Answer Here: Typeequationhere. 2. Using your equation for time above, substitute your values and find the actual time the car is in the air after it leaves the ramp. Answer Here: Typeequationhere. 3. Find the average horizontal distance dx the car travels before striking the floor. Show your work in the space below and be sure to include the proper number of significant digits. Answer Here: Typeequationhere. 5

4. Knowing the average horizontal distance dx your car traveled before striking the floor and the equation below, find the speed v of the car in m/s as it leaves the track. Answer Here: Typeequationhere. 5. Using the equation for kinetic energy below, find the kinetic energy in Joules of the car is it leaves the end of the track. Answer Here: Typeequationhere. 6

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6. Using the equation for potential energy, find the potential energy in Joules of the car when it is at the top of the ramp at the height h. PE = mgh Typeequationhere. 7. According to the law of conservation of energy, the total energy of a system is neither created nor destroyed. The total energy of the car at the top of the ramp (point A) is its potential energy, and the total energy of the car at the end of the track (point B) is its kinetic energy. A. According to your calculations, is the potential energy of the car at point A greater than, less than, or equal to its kinetic energy at the point B? Calculate any loss or gain in energy between points A and B. Answer Here: Potential Energy is greater than kinetic energy. The energy was lost due to friction and was dissipated in the form of heat (between the wheels and the track) B. If the energy at point A is greater than at B, where did the energy go? If the energy at point B is greater than at A, where did it come from? Answer Here: When the experiment started, the car was at point A, at some height grater than at B. That’s the potential energy, the today energy since the car started at rest. If there was no friction, then the energy at B (kinetic energy) would equal the potential energy at A. But due to friction, energy was lost, and the total energy at B is less than at A. No energy was created, but some of it was lost as heat. 8. Using the equation below, find the percent of total energy lost or gained between points A and B. %Energy Lost = Energy at A − Energy atB Energy at A x 100 7

= 73.4% 9. Calculate the energy lost or gained per centimeter of track between points A and B. Energy lost ∨ gained per cm = Energylost ∨ gained L Typeequation here. PART II: HEIGHT TO COMPLETE A LOOP Any time the car is in contact with the track, the car is losing energy. In question 9 in the Analysis section of Part I above, you calculated the energy lost per centimeter of track. You will use this value to help you predict the minimum height h from which to release your car in order for it to complete your loop without losing contact with the track. 1. Should the height H of the loop be greater than, less than, or equal to the initial height h of the car at point A? Explain your answer. Answer Here: the height should be less than because the car has potential energy at the start, but it loses energy through contact with the track, so it has less energy to go through the loop 2. Measure the maximum height H in centimeters for your loop. H= 0.33 m 8

3. Calculate the potential energy PE of the car at the measured height H of the loop. Be sure to use meters for the height so that the units for potential energy will be Joules. Answer Here: pe = mgh 0.039*9.8*0.33 4. Using your value for the energy lost per centimeter of track, calculate the energy lost on the track between the bottom of the loop and the top of the loop. (Hint: the amount of track between the bottom of the track and the top is half the circumference of the loop, and circumference = •diameter.) π Answer Here: 0.0012 J Elost = ½ π * 0.333 * 0.0023= 0.0012 J 9

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5. Calculate the minimum velocity for the car when it reaches the top of the track in order for the car to complete the loop without loosing contact with the track. Remember, the centripetal force must equal the gravitational force at this point. F g = F c 1.277 mg = m v 2 r rg = v 2 v = √ gr 6. In order for the car to make it around the loop at the top without losing contact with the track, it must have this minimum or critical velocity. Calculate the kinetic energy for this minimum velocity at the maximum height H. KE T = 0.032 J Show Work Here: Typeequationhere. 7. The kinetic energy at the bottom of the loop must be enough to convert to potential energy at the top of the loop, and also enough to overcome energy losses around half of the loop plus enough to supply the kinetic energy required to complete the loop. Add these values together to determine the required kinetic energy of the car just as it enters the loop. On the diagram below, label the values for kinetic energy at the top (KE top ), the energy lost on the track between the bottom and the top (E lost ), and the potential energy at the top (U gT ). 8. Add these values to obtain the required KE at the bottom or beginning of the loop. Show work Here: Typeequationhere. Required KE bottom = 0.16 J 10

KEbottom= 0.032 + mgh + 0.0023 (1/2h * π ) 11

9. Either use the measured length of the track or calculate its total length. Assuming the length of track before the car reaches the loop is h/2, π a quarter circle with radius h, use this equation to calculate the minimum height from which the car must be released to complete the loop. U g − E lost on track = KE beginning botton ofloop mgh − ( πh 2 ∙ 8.7 x 10 − 3 J m ) = K E beginningbottomofloop ShowWork Here : ( πh 2 ∙ 8.7 x 10 − 3 J m ) = 0.16 mgh= 0.16 + πh 2 ∗ 8.7 x 10 − 3 J m 0.039*9.8*h=0.16 + π ( 0.333 ) 2 * 8.7 x 10 − 3 0.3822h = 0.16455 H= 0.4305 m CONCLUSION QUESTIONS 1. In terms of energy gains and losses, what are some things a rollercoaster design engineer must take into consideration when designing a rollercoaster? Answer Here: Things rollercoaster design engineers must consider is how the ride affects the passengers. They must calculate the changes in momentum from their original source of acceleration since the human body can only withstand a certain amount of force or acceleration. 2. If you chose to add a second hill in the track after the loop, what are some of the things you would need to consider to determine the height of the hill? Answer Here: Since kinetic energy is converted into potential energy, every increase in height follows by a corresponding decrease in speed. Since potential energy is converted into kinetic energy, every decrease in height is accompanied by an increase in speed. 3. Would a passenger feel “heavier” as she passes the bottom of the loop or the top of the loop? Explain your answer, and draw and label vector arrows representing the forces acting on a passenger at the bottom of the loop and at the top of the loop. (Hint: There are two forces acting on the passenger at both the top of the loop and the bottom.) 12

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Answer Here: ft= reaction by surface w= weight of tension Fc= net centripetal force “Ft “ is the net weight felt by the passengers so by comparing we can say that “ft” is greater at the bottom than at the bottom than at the top so passengers will feel heavier at the bottom than at the top. 4. If we say that the passengers are experiencing an acceleration of one “g”, we mean that they are experiencing an acceleration of 9.80 m/s 2 . If a vertical rollercoaster loop has a radius of 10.0 meters and the speed of the rollercoaster at the bottom of the loop is 20.0 m/s and 5.0 m/s at the top, how 13

many g’s of acceleration do the passengers experience at the following points on the loop? Recall that the equation for finding the centripetal acceleration is a c = v 2 r M= mass of body A= acceleration as experienced by body V= velocity of passengers π = radius of loop A. at the bottom of the loop Show work and answer here: Typeequationhere. Nearly 5 times the value of g, passengers experience 5g acceleration B. at the top of the loop? Show work and answer here: Typeequationhere. 14

A cannot be negative which means that at the top the weight will provide the necessary centripetal required and Ft will be zero, the acceleration experienced will be 0 , body will be under weightlessness, so a top = 5. Using the values for the energy of the car you obtained in Analysis questions 4 and 5 above, find the speed of the car at points 1, 2, and 3, as shown on the diagram below. Show work here: 15 V 3 = 3.16 m/s V 2 = ________m/s V 1 = ________m/s

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6. On the diagram above, draw and label vector arrows representing the forces acting on a passenger at the bottom of the loop (1), halfway between the bottom and the top (2), and the top of the loop (3). 7. Calculate the centripetal force acting on the car at points 1, 2, and 3. Remember that the centripetal force acting on the car at any point on the circular track is the net force acting on the car. Show work here: mv 2 r 16

Rollercoaster Lab.docx 11

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