Chapter 9: Circular Motion

Transcription

1 Text: Chapter 9 Think and Explain: 1-5, 7-9, 11 Think and Solve: --- Chapter 9: Circular Motion NAME: Vocabulary: rotation, revolution, axis, centripetal, centrifugal, tangential speed, Hertz, rpm, rotational speed, linear speed Equations: f = 1 T v = 2πr T a c = v2 r F c = mv2 r Key Objectives: Concepts determine the directions of the velocity, acceleration and net force as an object travels in a circle. compare and contrast the terms rotation and revolution. identify the individual forces that are actually causing circular motion. explain what is meant by the term centripetal. Problem Solving convert between frequency and period convert between rpm and Hz and between minutes and seconds calculate the missing variable between speed, radius and time. calculate the missing variable between centripetal acceleration, speed and radius. calculate the missing variable between centripetal force, mass, speed and radius. determine the net force and the individual forces acting on an object going in a circle with a constant speed and constant radius. determine the net force and the individual forces acting on an object going in a vertical circle with a constant speed and constant radius. calculate the minimum speed needed for an object to just make a loop-the-loop

2

3 Period & Frequency NAME: Two seemingly simple terms often cause confusion for students because they are very similar. These are Period and Frequency. The purpose of this sheet is to give you the definitions of these terms and get you comfortable recognizing and converting between them. Period Symbol Defintion Units Frequency Period: 1 min = seconds & 1 second = minutes Frequency: 1 Hz = rpm & 1 rpm = Hz f = 1 T & T = 1 f Fill out the missing numbers in the chart below: Period Frequency seconds minutes Hz RPM a) 60 s b) 2 min c) 20 s d) 0.25 min e) 2 Hz f) 2 rpm g) 180 rpm h) 10 Hz i) 500 min j) 4.3 rpm side 1

4 Period & Frequency NAME: Questions 1. For each of the following, tell whether I am giving you a period (T) or a frequency (f): a. A car takes 24 seconds to go around a circle once. b. A kid is spun around at 3 revolutions per minute. c. A washing machine is spinning at 45 rpm. d. A cd rotates once every seconds. e. A wheel goes around at a rate of 3.5 Hz. 2. A runner does 4 laps around a track in 120 seconds. a. What is the period of the runner in seconds? b. What is the period of the runner in minutes? c. What is the frequency of the runner in Hz? d. What is the frequency of the runner in rpm? 3. What is the frequency of a tire that takes seconds to rotate once? 4. What is the period of a record that spins at 33.3 rpm? 5. What is the period of something that rotates at 20 Hz? 6. What is the frequency of a kid walking around in a circle once every 5 minutes? 7. A car takes 330 seconds to make one lap around a track. What is its rpm? 8. A Merry-go-Round rotates 3.5 times every minute. How many seconds does it take to go around once? Answers: 1. a) T b) f c) f d) T e) f 2. a) 30 s b) 1/2 min c) Hz d) 2 rpm 3) 40 Hz 4) 0.03 min 5) 0.05 s 6) 0.2 rpm 7) 0.18 rpm 8) 17.1 s side 2

5 Round and Round We Go NAME: Perform the following experiments in any order. Answer the questions in the spaces provided. Please leave equipment at each lab station for the next person. 1. Pennies in orbit Balance a penny on the end of the hanger hook as shown. With some practice, you can swing the hanger in circles while the penny remains in place. (For an easier orbit, use the cardboard hanger as shown in the second figure). Q: Why doesn t the coin fall when it is at the top of its circular path? 2. Cup in orbit Place some water in the paper cup and center the cup on the platform. Begin swinging the platform back and forth until you are able to swing the platform and cup in complete circles. (Be sure to wipe up any spills.) Q: Why doesn t the cup fall off the platform when it is at the top of its circular path? 3. There s a Hole in My Cup Using a bottomless cup, try to pick up a marble. Q: What technique did you use and why did it work? 4. Accelerometer The level of the water in the accelerometer shows you the direction of the acceleration. If it is not accelerating, the water level is flat, and if the water level is tilted, then it is accelerating, as shown below: a = 0 a a Because it is so thin, it does a nice job of showing you only one component of the acceleration. Sit on the stool, hold the accelerometer out away from your body with one hand and give yourself a spin. While spinning, use the accelerometer to determine the direction of the acceleration. Q: In what direction is your hand accelerating? 5. Ball on a String Swing the ball on the string in circles above your head. BE CAREFUL NOT TO HIT ANYONE! Q: What would happen if you were to let go of the string? Try it! PLEASE BE CAREFUL NOT TO HIT ANYONE. Describe the motion of the ball after the string is released. Why does this occur? (HINT: Remember Newton s First Law of Motion?) side 1

6 Round and Round We Go NAME: 6. Stairwell Demo Obtain permission from your teacher to briefly leave the room. Carefully run up the stairs from the first to second floor in the stairwell by Mr. Bradford s room (262W). Note what you must do in order to turn the corner at the landing. Q: Could you have made the turn without holding on to the railing? Describe the direction of the force your arm exerts on your body as you make the turn. In which direction would your body travel if you attempted to make the turn at high speed without the use of the railing? 7. Ye Olde Bucket of Water Swing a bucket, about 1/3 full of water, over your head in a circle. (Be sure to wipe up any spills.) Q: Why does the water remain in the bucket? What would happen if the bucket were to stop directly over your head? Q: You know that gravity imparts a downward acceleration of 10 m/s 2 to the water when the bucket is at the top. If the water stays in the bucket at the top, what can be said about the acceleration of the bucket at the top? 8. Paper Plate Puzzle If you roll a marble around the inside of the plate s rim, in which direction will the marble go when it gets to the cut edge? Will it continue to curve inward, go straight ahead, or curve outward? Try it! marble??? Q: Explain your observations: plate CONCLUSIONS: 1. When an object is moving in a circle with constant speed, is it accelerating? Explain your answer. 2. The direction of the acceleration of an object moving in a circle is. 3. To produce this acceleration, there must be a. 4. Therefore, whenever we see an object moving in a circle, we know there must be a acting on the object. The direction of the net force is always to the motion of the object and directed to the of the circle. 5. If an object were traveling in a circle, and the force causing this motion were to somehow disappear, what would happen to the object? side 2

7 1. Define the following terms. Linear speed Rotational speed Period Frequency Hertz RPM Linear Speed NAME: 2. Joanne puts her favorite disc in the CD player. If it spins with a frequency of 1800 rpm. a. What is the frequency of rotation in Hz? b. What is the period of rotation? 3. Hamlet, a hamster, runs on his exercise wheel, which turns around once every 0.5 s. a. What is the frequency in Hz of the wheel? b. How many rpm is that? 4. You are walking in circles with a radius of 150 meters in a big field. It takes you 5 minutes to go around once. a. What is your frequency in rpm? b. What is your frequency in Hz? c. How far do you travel in going around once? d. What is your linear speed? 5. A sock stuck to the inside of the clothes dryer spins around the drum once every 2.0 s at a distance of 0.50 m from the center of the drum. a. What is the sock s linear speed? b. If the drum were twice as wide but continued to turn with the same frequency, would the linear speed be faster than, slower than or the same as your answer to part a? side 1

8 Linear Speed NAME: 6. Charlotte twirls a round piece of pizza dough overhead with a frequency of 60 revolutions per minute. a. Find the linear speed of a piece of pepperoni stuck on the dough 10 cm from the pizza s center. b. In what direction will the pepperoni move if it flies off while the pizza is spinning? Explain. 7. A car has a linear speed of 12 m/s while it drives around in a circle. The radius of the circle is 50 meters. a. How many seconds will it take the car to go around once? b. What is the frequency in Hz of the car? 8. A record player works by spinning a record at a constant rate of 33.3 rpm. A needle then floats in a groove that spirals around the record, moving from the edge of the album to the middle of the album. (The needle picks up the vibrations from the groove, and turns it into an electrical signal.) a. How many seconds will it take for one complete rotation? b. What is the linear speed of a point on the edge of the record with a radius of 15 cm? c. What is the linear speed of a point in the middle of the record with a radius of 5 cm? *9. A CD player works by spinning the CD and having a small laser track a groove etched into the CD. (The laser looks at little pits that are in the groove, and sends a digital signal back to the processor.) The laser always moves with a constant linear speed that depends on the player, but let s say the linear speed is 12 m/s. a. When the laser is on the inside of the CD with a radius of 5 cm, what is the frequency of the spinning CD? b. When the laser is on the outside of the CD with a radius of 10 cm, what is the frequency of the spinning CD? Answers: 2. a) 30 Hz b) s 3. a) 2 Hz b) 120 rpm 4. a) 0.2 rpm b) Hz c) 942 m d) 3.1 m/s 5. a) 1.57 m/s b) twice 6. a) m/s b) tangent to circle 7. a) 26.2 s b) Hz 8. a) 1.8 s b)0.52 m/s c) m/s 9. a) 38.2 Hz b) 19.1 Hz side 2

9 Basic Notes A. What does the term "centripetal" mean? Centripetal Acceleration Notes NAME: B. If you go in a circle with a constant speed, why are you accelerating? C. What is always true about the direction you move when you go around in a circle? D. In which direction are you accelerating when you go around in a circle? E. What is the equation that relates centripetal acceleration, speed and radius? v F. The diagram to the right represents something going in a circle with a constant speed and constant radius. At one point, the velocity and acceleration are shown. Draw appropriate vectors to represent the velocity and acceleration for the other points on the circle. a Questions 1. How does the direction of your velocity compare to the direction of your acceleration if you are going in a circle with a constant speed? 2. If you tried to go around a circle twice as fast (but same radius), what has to happen to your acceleration? 3. If you tried to go around a circle with twice the radius (but the same speed), what has to happen to your acceleration? 4. If somehow your acceleration was always perpendicular to your velocity, describe your motion. 5. For each of the following amusement park rides, describe the direction of your acceleration: a. On a Ferris Wheel, when you are at the highest point. b. On a Ferris Wheel, when you are at the lowest point. c. On a loop-the-loop coaster, when you are at the highest point. d. On a loop-the-loop coaster, when you are at the lowest point. e. On the Turkish Twist. side 1

10

11 Centripetal Acceleration Concepts A. If you are going in a circle with a constant speed, why are you accelerating? NAME: B. If you are going in a circle with a constant speed, in what direction do you accelerate? C. If you are going in a circle with a constant speed, describe the direction of your velocity. Calculations 1. A car is traveling in a circle with a radius of 20 meters. a. If it has a speed of 5 m/s, what is the acceleration of the car? b. If it has a speed of 10 m/s, what is its acceleration? 2. A plane is flying at 125 m/s when it begins to travel in a circle. If its centripetal acceleration is 2 m/s 2, what is the radius of the circle? 3. A girl is sitting on a merry-go-round 2 meters from the center. a. If she has an acceleration of 1 m/s 2, how fast is she going? b. If she has an acceleration of 2 m/s 2, how fast is she going? 4. A person is driving in a circle with a centripetal acceleration of 2 m/s 2. a. What would be the acceleration if they went twice as fast, but kept the radius the same? b. What would be the acceleration if they went three times as fast, but kept the radius the same? c. What would be the acceleration if they doubled the radius, but kept their speed the same? d. What would be the acceleration if they tripled the radius, but kept their speed the same? 5. A car is traveling in a circle of radius 15 meters. It takes 9 seconds to go once around the circle. What is the centripetal acceleration? (Hint: Find the speed first.) side 1

12 Centripetal Acceleration NAME: 6. A ball is swung on a string in a circle of radius 1.3 meters. If the centripetal acceleration of the ball is 15 m/s 2, how long does it take the ball to go around once? (Hint: Find the speed first.) 7. While flying in circles, a plane has a centripetal acceleration of 5 m/s 2. If the radius of the turn is 8000 meters, how many seconds does it take to go around once? (No more hints!) 8. A person is spinning on the Turkish Twist, which has a radius of 5 meters. If it takes 2.5 seconds to go around once, what is the centripetal acceleration of the person? 9. A ball on the end of a string is being spun in a circle of radius 2.3 meters. It is spinning at a rate of 45 rpm. What is the centripetal acceleration of the ball? 10. A person on a 10 meter radius Ferris wheel is rotating with a centripetal acceleration of 4 m/s 2. What is the rate of rotation in rpm? Answers: 1. a) 1.25 m/s 2 b) 5 m/s 2 2) 7800 m 3. a) 1.4 m/s b) 2 m/s 4. a) 8 m/s 2 b) 18 m/s 2 c) 1 m/s 2 d) 0.67 m/s 2 5) v = 10.5 m/s & a = 7.3 m/s 2 6) v = 4.42 m/s & t = 1.85 s 7) v = 200 m/s & t = 251 s 8) v = 12.6 m/s & a = 31.6 m/s 2 9) v = 10.8 m/s & a = 51 m/s 2 10) v = 6.32 m/s & T = 9.93 s & f = 6.04 rpm side 2

13 Lab 9-1: Centripetal Force NAME: Purpose: Whenever an object moves in a circle with constant speed and radius, the net force on the object is always directed to the center of the circle. The net force in this situation is given the special name, centripetal force, which simply means "center-seeking" force. Centripetal forces depend on an object's mass, speed, and radius of the circular path. In this lab, you will determine how centripetal forces depend on the speed of an object. Materials: 1 hanger 1 glass tube 1 rubber stopper 1 string (~1 m) 1 stop watch slotted masses (total of 250 grams) Procedure: 1. Find the mass of the rubber stopper, record it in the data table, and then set up your apparatus as shown in the diagram below. tube 0.75 m pen mark on string (keep this mark level when spinning) hanger with slotted masses rubber stopper 2. Adjust the length of the string so that there is 0.75m from the glass tube to the middle of the stopper. Using a pen or marker, make a small mark on the string just where it comes out the tube. (This will give you a reference point to keep the radius constant at 0.75 m while spinning the stopper.) 3. Without any additional masses on the hanger, practice spinning the stopper. You need to be able to spin the stopper in a horizontal circle over your head and keep the piece of tape at the same distance below the glass tube. Be careful not to hit any passersby while you are spinning the stopper! 4. Without any additional masses on the hanger, spin the stopper. When you are ready, time how long it takes for the stopper to make 20 revolutions. (This is easier if someone counts and someone else uses the stop watch.) Record your results. 5. Add 50 grams to the hanger, and repeat step #5. Fill out the data table, each time adding an additional 50 grams to the hanger. Calculations: 1. Calculate the circumference of the circle that the stopper always traveled in. Record in the data table. 2. Calculate the speed of the stopper for each trial and record in the data table. Show your first calculation here: 3. Calculate the square of the speed of the stopper for each trial and record in the data table. Show your first calculation here: side 1

14 Lab 9-1: Centripetal Force NAME: Data: Mass of rubber stopper = kg Radius of circular path = 0.75 m Note: While doing the lab, the only data you need to take is the third column of the data table (Time for 20 revolutions). The rest of the table is calculated. Mass hanging (kg) Weight hanging F c (N) Time for 20 revolutions (s) Period of 1 revolution (s) Circumference of circle (m) Speed of stopper (m/s) v 2 (m/s) Graph: Make a graph of F c vs. speed 2. Make sure you can see the origin in the graphs. Make sure the graph has a title, labels, units and the regression line. Check with your teacher and if it is ok, print the graph. Conclusion: 1. What is the equation that relates centripetal force and speed for your experimental setup? 2. Show that the units of the slope of the straight line reduces to kg/m. (Hint: what is a N?) 3. Divide the mass of your stopper by the radius of the circle. 4. What is the physical significance of the slope of this equation? 5. Define the following terms: a. Revolution b. Period c. Net Force d. Centripetal Force side 2

15 Centripetal Force NAME: Concepts A. In what direction are you accelerating if you are moving in a circle with a constant speed? B. If you are accelerating to the left, in what direction is the net force on you? Generalize this statement for any acceleration. C. In what direction is the net force on you if you are moving in a circle with a constant speed? D. Centripetal Force is just another name for the force acting on something when it is doing what? E. If there is no net force on you, can you move in a circle at constant speed? Explain. Calculations 1. A 1500 kg car is traveling in a circle with a 12 meter radius and a centripetal acceleration of 3 m/s 2. a. How fast is the car traveling? b. What is the centripetal force on the car? c. Where does the centripetal force come from? 2. A 75 kg person is on a Ferris Wheel of 5 meter radius that is rotating, If the person has a speed of 2 m/s, a. What is the centripetal acceleration of the person? (Give magnitude and direction.) b. What is the centripetal force on the person? (Give magnitude and direction.) 3. An airplane of mass 15,000 kg is traveling with a speed of 75 m/s. If turns with a radius of 200 meters, what is the centripetal force needed to let the airplane turn? 4. There is a 1700 kg car traveling in a circle with a radius of 15 meters a centripetal force of 5000 N acting on it. How fast is the car going? side 1

16 Centripetal Force NAME: 5. A 75 kg person is running in a circle. There is a centripetal force of 50 N acting on the person, and they are running at 3 m/s. What is the radius of the circle? 6. An airplane of mass 15,000 kg is traveling with a speed of 75 m/s. It turns with a radius of 2000 meters. a. What is the centripetal acceleration of the plane? b. What is the centripetal force on the plane? c. What is the net force on the plane 7. A 2500 kg car is driving around a circle with a radius of 15 meters. There is a centripetal force on the car of 10,000 N. a. How fast is the car going? b. What is the net force on the car? c. If there was no friction, what would happen to the car? 8. A 5 kg bag is swung in a circle at a speed of 3 m/s. There is a centripetal force of 20 N acting on the bag. a. What is the radius of the circle? b. What is the centripetal acceleration of the bag? Answers: 1. a) 6 m/s b) 4500 N c) friction between tires and road 2. a) 0.8 m/s 2, to the center b) 60 N, to the center 3) 422,000 N 4) 6.64 m/s 5) 13.5 m 6. a) 2.81 m/s 2 b) 42,200 N c) 42,200 N 7. a) 7.75 m/s b) 10,000 N c) car would fly off tangent with a constant velocity 8. a) 2.25 m b) 4 m/s 2 side 2

17 Vertical Circles 1. Imagine you (mass of 70 kg) are riding a Ferris Wheel with a radius of 15 meters. It takes 17 seconds to make one complete rotation. Some warm up questions first: a. What is the frequency of your rotation in Hertz? NAME: A b. What is your linear speed? D B c. What is your acceleration? (Give magnitude and direction.) d. What is the net force on you? (Give magnitude and sketch directions in the diagram.) C Now for some harder questions: e. When you are at your lowest position on the ride (C), there are two forces acting on you. What are they? Draw a force diagram. f. What is your weight? (Remember, for this problem, you have a mass of 70 kg.) g. You know the net force (from part d above), and you know the force of gravity on you (from part f) so what must be the normal force acting on you? h. What is the normal force acting on you when you are at the highest position on the Ferris Wheel (A). (First think about the two forces acting on your body, and then think about the net force on you.) i. Why are your answers to g and h different? Why does the normal force on you change? j. If you have ever ridden a Ferris Wheel, you feel a little heavier at the bottom and a little lighter at the top. Why? 2. Now you (mass 70 kg) are riding a fast Ferris Wheel (radius 11 meters). At the highest point, the normal force on you is only 300 N. a. What must be the net force on you? b. What is the normal force when you are at the lowest point? c. What is your linear speed? d. How many seconds will it take go around once? side 1

18 Vertical Circles NAME: 3. A bag of books has a mass of 10 kg. A happy physics student is swinging the bag in a vertical circle of radius 0.90 meters. The student is swinging the bag with a speed of 10 m/s. a. What is the net force on the bag of books? In which direction does it point? b. How much force must the student provide when the bag is at the top of the circle? c. How much force must the student provide when the bag is at the bottom of the circle? d. Why would these numbers be different? 4. The same student with the same books from the previous problem is now getting tired. a. What is the minimum speed with which the student must swing the books in order for the books to stay in the bag at the top of the swing? b. What force must the student provide at the top of the swing? c. What force must the student provide at the bottom of the swing? (Assume a constant speed for the books.) 5. Still the same student and same books. If the maximum force that the student can provide is 250 N, what is the maximum speed that the student can swing the books at? (Be careful on this. Think about the force diagram on the books and where the student will need to pull with the most force.) Answers: 1. a) Hz b) 5.54 m/s c) 2.05 m/s 2, to the center d) 143 N, to the center e) weight & normal f) 700 N g) 843 N h) 557 N 2. a) 400 N b) 1100 N c) 7.93 m/s d) 8.72 s 3. a) 1111 N, always to the center b) 1011 N c) 1211 N d) gravity is helping you at the top, but fighting you at the bottom 4. a) 3 m/s b) 0 N c) 200 N 5) 3.67 m/s side 2

19 Purpose: Lab 9-2: Hot Wheels NAME: 1. To define and determine the efficiency of a Hot Wheels track. 2. To determine the height from which a Hot Wheels car must be released to have it just make it around the loop-the-loop track. Discussion: This lab combines the ideas of centripetal force and conservation of energy. You will set up a loop-the-loop with Hot Wheels track and calculate the minimum height from which to release a Hot Wheels car so that it safely completes the loop. In order to do this, you must first derive an expression for the minimum height from which to release the car so that it just makes the loop-the-loop in an ideal case. Then you will determine the energy efficiency of your Hot Wheels set up, and combine the results to determine the actual release point. Part 1: Determining the efficiency of your Hot Wheels track. A Hot Wheels car is released from point A. It reaches point B on the opposite side. height 1 B height 2 If a Hot Wheels car were placed on a U-shaped track and released, it would ideally rise to the same height on the opposite side of the track. The potential energy of the car at the top of the track would change into kinetic energy at the bottom of the track. The kinetic energy would then change back into potential energy as the car goes up to the same height on the opposite side. (No energy would be lost to any other transformation.) In the lab, however, there are a number of other energy transformations that take place, and some of the car's original potential energy is "lost" to these other transformations; the car will not rise to the same height. We will define the efficiency of the Hot Wheels set-up as the ratio of how much potential energy the car has on the far side of the track to how much potential energy the car had originally. In symbol form: Efficiency = PE final PE original Procedure: l. Set up the Hot Wheels track as shown in the diagram above. Secure the track with tape, books, braces etc. so that the track does not move when the car is released. 2. Release the hot wheels car from starting point A. This height (h 1 ) and the height to which the car rises (h 2 ) on the other end of the track are to be recorded. Data: Trial h 1 (cm) h 2 (cm) average side 1

20 Lab 9-2: Hot Wheels Part 2: Determining the minimum height from which to release the Hot Wheels car NAME: Discussion: The hot wheels car is going to come down a hill with enough speed so that it will be able to go through a loop and just barely leave the track at the top of the loop as it continues through the rest of the loop. So what is the minimum height? If the track were 100% efficient, we know that the minimum height from which to release the car is just two and one half times the radius of the loop. h ideal = 2.5r The Hot Wheels track is not 100% efficient, however. The car must be released from a higher point than given by the above equation because some energy is lost. Procedure: 1. Do Questions 1 to 3 (below) to determine your predicted minimum height. 2. Set up the loop-the-loop. 3. Put the Hot Wheels car on the track at the calculated height, and let it go! 4. Experimentally determine the minimum height to see how close your prediction was. Questions: 1. From Part 1, what is the ideal minimum height so the car just makes the loop? 2. From Part 2, calculate the efficiency of your track. (Think of it as a fraction - how much energy did the car keep going from one side to the other.) 3. Calculate the actual height from which you should release the car so that it just makes the loop. 4. How close was your predicted minimum height to the actual minimum height? 5. Why is your set-up not 100% efficient? Specifically, what happened to the energy that the car "lost" in going from point A to point B? 6. What is the normal force on the car at the top of the loop if it just barely makes the loop? 7. Why doesn't gravity pull the car off the track? side 2

21 Lab 9-2: Supplement NAME: Derivation of formula to find minimum height. We need to find the minimum height from which to release the Hot Wheels car so that it just barely completes a loop-the-loop. We will assume that there is no friction in this derivation. While the car is in the loop-the-loop, it is traveling in a circle. The net force on the car must be directed towards the center of that circle at all times; there is a centripetal force acting on the car! Looking at the very top of the loop-the-loop, while the car is upside down, the centripetal force comes from two individual forces: the weight of the car and the Normal force of the track pushing on the car. (see diagram below) Normal force of track pushing down on the car through the wheels (dotted arrows) Weight of car pulling down (fat arrow) Hot Wheels Loop-the-loop (with car at top of loop) The normal force and the weight together act as a centripetal force to keep the car moving in a circle. The weight of the car is constant. The normal force, however, depends on how fast the car is traveling. If the car is going very fast, the normal force will be very large. The slower the car is moving, the less the normal is. The smallest value the normal force can have is zero. This means the track is no longer pushing on the car, and the only force acting on the car is the car's own weight. In order for the car to just make the loop-the-loop, the normal force must be zero, and the weight of the car is the centripetal force making the car travel in a circle. In equation form: weight = centripetal _ force mg = mv2 r After doing a little algebra, this equation becomes v 2 = gr (equation 1) Equation 1 tells us how fast the car must be going in order to just stay on the track of a given radius, r, but it does not tell us how high the car must be released on the Hot Wheels track. We need another equation that relates speed and height; let's look at kinetic and potential energy. A Hot Wheels car at release point, A ha B r final height, hb (= twice radius of loop) side 1

22 Lab 9-2: Supplement NAME: By placing the car at a certain height, we are giving it gravitational potential energy. When we release the car, its potential energy is transformed to kinetic energy. If we assume that there is no friction, the kinetic energy of the car at any point on the track is just equal to the change in its potential energy (because it is at a lower point on the track.) Because of the conservation of energy, the potential energy the car has at the top of the Hot Wheels track is equal to the sum of it kinetic energy plus its lower potential energy at the top of the loop-theloop. In equation form: Potential Energy A = Potential Energy B + Kinetic Energy B PE A = PE B + KE B mgh A = mgh B mv2 B Now we can substitute in equation 1, and simplify the energy equation. mgh A = mgh B m(gr) h A = h B r We are almost done! The height of the car at the top of the loop-the -loop, h B, is just twice the radius of the loop (h B =2r), so we can rewrite the equation as h A = (2r) r so the minimum height, h A, is simply h A = 2.5r So the minimum height needed for the car to just make the loop is two and a half times the radius of the loop! side 2