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 = 2!r 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.! compare and contrast the terms centripetal and centrifugal.! explain why centrifugal forces are not real forces. Problem Solving! convert between frequency and period! convert between rpm and Hz! 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.! apply the conservation of energy to an object falling on the end of a string or a sliding along a circular track and determine the net force and applied forces on the object. (i.e. Vertical Circle lab)

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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 Fill out the missing numbers in the chart below: Period Frequency seconds minutes Hz RPM 60 s 2 min 20 s 0.25 min 2 Hz 2 rpm 120 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 Rotation and Revolution Practice NAME: 1. Define the following terms. Linear speed Rotational speed Period Frequency Hertz RPM 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

6 Rotation and Revolution Practice 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) 76.4 Hz side 2

7 Centripetal Force I 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? NAME: 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 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? 5. 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? b. What is the centripetal force on the person? c. In what direction is the person accelerating? d. In what direction is the centripetal force? side 1

8 Centripetal Force I NAME: 6. 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? 7. 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 going? 8. 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? 9. 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? 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) 6 m/s b) 4500 N c) friction 5. a) 0.8 m/s 2 b) 60 N c+d) to the center 6) 422,000 N 7) 6.64 m/s 8) 13.5 m 9. a) 8 m/s 2 b) 18 m/s 2 c) 1 m/s 2 d) 0.67 m/s 2 side 2

9 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

10 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

11 Centripetal Force II NAME: 1. 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? 2. 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? 3. 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? 4. 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? 5. 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? 6. 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) 7.3 m/s 2 2) 1.85 s 3) 251 s 4) 31.6 m/s 2 5) 51 m/s 2 6) 6.04 rpm side 1

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13 Centripetal Force III NAME: 1. 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 2. 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. Where does the centripetal force on the car actually come from? 3. 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? 4. 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? 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? side 1

14 Centripetal Force III NAME: d. Why would these numbers be different? e. What is the frequency of the swinging bag of books? 5. The same student with the same books from the previous problem is now getting tired.. a. What is the minimum speed the student must swing the books with 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? 6. 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) 2.81 m/s 2 b) 42,200 N c) 42,200 N 2 a) 7.75 m/s b) 10,000 N c) friction between tires and road 3 a) 2.25 m b) 4 m/s 2 4 a) 1111 N b) 1011 N c) 1211 N e) 1.77 Hz 5 a) 3 m/s b) 0 N c) 200 N 6) 3.67 m/s side 2

15 Purpose: Lab 9-2: Vertical Circles 1. To calculate the centripetal force on a mass at the end of a string. 2. To measure the centripetal force on a mass at the end of a string. Equipment: force probe, stand string mass meter stick Procedure: 1. Hang the mass from the string and attach it to the force probe as shown. Make the string as long as possible, but make sure that the pendulum can swing freely. Also make sure that the force probe is vertical. See diagram. 2. Record the radius of the pendulum you have just made, from the top point where it swings to the middle of the mass. With the mass simply hanging, record the tension in the string, which is simply the weight of the mass. Record the initial height of the mass (from the table to the middle of the mass.) 3. Pull the mass back so that it makes a 30º to 45º mass force probe NAME: angle. Measure the maximum height of the mass (from the table to the middle of the mass.) 4. Making sure that someone is holding the stand steady, start collecting data and let go of the mass. Record the tension in the string when the mass gets to its lowest point. (This will be the maximum force read by the force probe.) 5. Repeat steps 3 and 4 two more times. You can either drop the mass from the same height every time, or try it from different heights. It s up to you. Just make sure you record your data. r h 1 h 2 hold stand steady Data: Radius of the pendulum: (m) Top height of mass Bottom height of mass bottom (m) (m) (N) Tension when mass is stationary: (N) trial 1 trial 2 trial 3 Calculations: For each of these calculations, show the equation you are using, and then show your work. You must do all these for each trial, but you only have to show your work for the first trial. Record your results in the table at the end of this section. 1. What is the potential energy lost by the swinging mass? 2. What is the kinetic energy of the mass at the bottom of its swing? 3. How fast is the mass going at the bottom of its swing? 4. What is the centripetal force needed so that the mass can be traveling is a circle with its calculated speed and radius? 5. What is the weight of the mass? side 1

16 Lab 9-2: Vertical Circles NAME: 6. Calculate the net force on the mass at the bottom of the swing. Results of Calculations: PE lost by mass (J) bottom of swing (J) bottom of swing (m/s) centripetal force needed (N) weight (N) tension in bottom * (N) net force on bottom (N) trial 1 trial 2 trial 3 * Just copy this from your data on the other side Conclusions: 1. Compare the net force on the mass to the centripetal force needed. Explain why your results either make sense or don t make sense. 2. If the mass were just hanging without moving, the tension in the string would simply be equal to the weight of the mass. When the mass is swinging, however, the tension in the string will always be greater than the weight of the mass when the mass is at the bottom of its swing. Explain this. side 2