Centripetal Motion and Conservation of Energy


 Kenneth Melton
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1 Lab #5 Centripetal Motion/Conservation of Energy page 1 Centripetal Motion and Conservation of Energy Reading: Giambatista, Richardson, and Richardson Chapters 5 (5.1, 5.2) and Chapter 6. Summary: The physics in this lab is actually used in the design of roller coasters. Some of the new ones use only gravity and centripetal forces to keep the roller coaster car on the track, even when the car is upside down. The equipment in this lab has been designed to simulate the standard roller coaster incline before and after the looptheloops. You will use a ball to simulate the roller coaster car. The car will be launched on one incline and must attain a certain critical velocity in order to loop around the track without falling off it. In this lab you will use your physics knowledge to calculate how high the car must start in order to complete the loop and then experimentally check your results. Also from experimental measurements with the car on the track, you will calculate how high up the other incline the car will rise. This is critical in designing roller coaster tracks, because you don t want the cars to fall off the track due to not planning for enough track after the looptheloop. Note: This is PreLab long, so start it early! Prelab Analysis: Figure 1 shows a schematic of the equipment you will use in the lab. You will release a ball at position A and, if A is high A L D enough, the ball will loop around, eventually rising to position E. The whole trick of this lab is to figure out how high A has to be in ha B order for the ball to roll all the way around the loop without falling off at D. To solve this Lab bench top C problem, we will need two physics principles: Figure 1. The looptheloop roller coaster. centripetal force and conservation of energy. v First you will calculate what velocity the ball must have to stay on the circular part of the track and then from that you will R work backward to determine how high A needs to be. Figure 2 a shows the velocity and acceleration vectors for a ball traversing the vertical loop in a counterclockwise direction. E 1. Write an expression for the centripetal acceleration a of the ball in terms of the ball s speed v and radius of motion R. [2 pts] 1 Figure 2. Circular motion. This centripetal acceleration is supplied by a combination of gravity and the normal force the track exerts on the ball. If the ball is traveling fast enough, the track supplies most of the force required to make the ball change direction and travel in a circle. If the ball is traveling too slowly, gravity will supply too much downward force and the ball will fall off the track. The slowest velocity at which the ball will go all the way around is when the track exerts no force on the ball at the top of the vertical loop. Then gravity supplies all of the centripetal force.
2 Lab #5 Centripetal Motion/Conservation of Energy page 2 2. a.) For this slowest velocity, show that v D = gr. To do this, begin by relating the force of gravity on the ball to the centripetal force the ball experiences when going around the loop. Then solve for vd from this equation. [4 pts] 2 b.) If the diameter of the loop is 30cm, how fast is this slowest velocity? [4 pts] 3 Next we need to find how high up the initial incline the ball has to start (point A) in order to attain this speed at the top of the vertical loop. For this, we use the conservation of energy law, which states that the ball s total energy at the release point (A) must be the same as its total energy at the top of the vertical loop, iiff there are no nonconservative forces acting (like friction). 3. What is the ball s total energy at: a.) the release point (A), if A is a height ha above the top of the lab bench and the ball is released with no initial speed? Leave your answer in terms of the ball s mass m, the acceleration due to gravity g, and the height ha. [2 pts] 4 b.) the top of the vertical loop (D), if only conservative forces act on the ball? Express your answer in terms of m, the ball s centripetal velocity (vd), and the loop s radius R. Don t forget that at the top of the vertical loop (height = 2R above the bottom of the incline), the ball has both kinetic and potential energy. [3 pts] 5 4. In reality, friction is present between the ball and the track, which means the ball rolls down the incline and around the vertical loop. This rolling motion has kinetic energy associated with it, meaning another term needs to be added to the kinetic energy. As you will learn later in the semester, this rotational kinetic energy is a( 1 /2)mv 2, where a 0.4 for the solid ball in this experiment. a.) Rewrite the total energy at point D (Question 3b) to include a. [3 pts] 6 b.) Since for now we are going to assume only conservative forces act between points A and B, use the conservation of energy law between these two points to show that the minimum height (hmin) required to make the ball go all the way around the loop is: h min = 2R a 2 ( ) Ê ( R) = Á 5 + a Hint: Set the two energy formulas from Questions 3a and 4a equal to each other and use the velocity at B from Question 2. [7 pts] 7 In this lab, the whole apparatus will be mounted on a stand that will be a height hstand above the top of the lab bench. Therefore the point A will be hmin + hstand above the lab bench top. Setting a = 0.4 is only approximate and ignoring friction is fine only in theory, but in the lab it cannot be disregarded. Thus you will need to determine experimentally both a and the amount of energy lost due to friction on the track. The a can be found from applying conservation of energy to points A and B on the lefthand incline. Ë 2 ˆ R
3 Lab #5 Centripetal Motion/Conservation of Energy page 3 D 5. This question takes you through the A L derivation of the equations you will B E need in the lab to experimentally determine a. Assume the ball starts ha at a height ha above the table and hb with no initial velocity at A. At Lab C point B, the ball has a velocity of vb Bench Top and is at a height hb above the table Figure 3. The apparatus showing the starting positions. (see Figure 3). a.) Write the total energy of the ball at points A and B. (Hint: Don t forget that the ball rolls down the incline, so the kinetic energy of the ball at point B has an additional term of ( 1 /2)mav 2.) [4 pts] 8 b.) If points A and B are close together, friction can be ignored, and energy will be conserved between these two points, making the experimental determination of a much simpler. Use energy conservation between A and B, to each show that: [6 pts] 9 2 2g v B = ( ) 1 + a ( ) h A  h B 6. In the lab, you will use the motion sensor to measure the final velocity of the ball as it passes point B for six different positions of B, to obtain a more accurate a. The table to the right contains such data. The track made a 15º angle with the lab bench top and the ball started with no initial velocity at point A. a.) Draw only the inclined part of the track and from the geometry come up with an equation that relates the height difference (ha hb) to the distance between points A and B ( L). [4 pts] 10 b.) Using your formula from (a) and Excel, calculate these heights (in m) for each distance given in the table. [9 pts] 11 v (m/s) L(cm) c.) Plot v B 2 vs (h A hb); fit a linear Trendline to it; print your data table and plot. [9 pts] 12 d.) From the slope of this line, determine a using the equation in Question 5b. [5 pts] 13 Lastly, if friction could be ignored over the whole length of the track, then when the ball reached point E it would rise to exactly the same height it started at (ha). However experimentally, the ball s highest final position is always lower than its starting position. Thus energy is not conserved over the whole track, as we assumed over the short distance from A to B. To determine the starting position that will allow the ball to just barely make it all the way around the loop without falling off the track at point D, we need to estimate the energy loss due to friction. Measuring the height difference between points A and E gives a reasonable estimate of these losses. If this height difference is hloss, total = ha he, then the ball will need to be started hloss, total/2 higher than the minimum position (Question 4b), assuming the frictional energy losses from A to D are approximately half those from A to E. 7. As an example of this, suppose the ball is released at 30 cm above the lab bench top (at A) and rises to a final height of 24 cm above the bench top (at E). How much above the
4 Lab #5 Centripetal Motion/Conservation of Energy page 4 minimum height distance (without losses) should you release the ball, for the ball to just make it all the way around the loop without falling off? (Hint: this is a very simple problem, so don t make it harder than it is.) [3 pts] Outline the lab following the format of Outline Format posted on the Electronic Reserves web page. (20 pts total) 15 Equipment to be used in this lab: r Aluminum looptheloop track r 1 plastic ball check this out from your TA (you are responsible for returning it) r 2 motion sensors attached to top of the track on its right and left sides. Experimental Procedure: 1. Safety r This lab is fairly benign, but the track does have sharp edges at about eye level. Keep an eye on (or rather off) the track. 2. Setup r Check that the lefthand motion sensor is connected to port 2 and that both sensors are mounted securely to the track. If they are not, notify your TA. r Double click on the Lab05Centripetal icon. A velocity (on y) versus distance (on x) graph should appear. Check that the Experiment Length is set to 4 seconds and the Sampling Speed is set to 20 samples/sec. 3. Determining a a.) Characterizing the Apparatus r Sketch the apparatus in your notebook. [1 pt] 16 r Choose a point A more than 0.4m along the track from the motion sensor (this is the minimum detection distance of the detector). Mark the side of the track with a small piece of tape. Make sure the tape does not overlap the groove where the ball will roll or it will slow the ball and all your measurements will be off. r Measure the height at A (ha) and record it on the sketch in your notebook. [2 pts] 17 r Place a piece of tape on the track 5cm down from A and the motion sensor. b.) Motion Sensor data collection between Points A and B r Place the ball at A. Hold a pen or pencil across the groove of the apparatus to prevent the ball from rolling down the incline. It is important that your hands and other body parts are not within range of the sensor when you release the ball or the motion sensor may not take data on the ball s motion. r In LoggerPro click on Collect and let go of the ball after you hear the second click from the motion sensor. r Click on Stop when the ball has passed the 5cm tape mark.
5 Lab #5 Centripetal Motion/Conservation of Energy page 5 r Print the velocity versus distance curve that appears on the screen and attach it to your notebook. [3 pts] 18 r Using the X = icon at the top of the LoggerPro screen and the pointer, find the velocities corresponding to three different B distances starting approximately 1cm from A and roughly 1cm apart. Record these velocities in a data table labeled Data for determining a. Label these velocities vb. [9 pts] 19 r Mark with tape the precise position of each of the three B points on the track. These will be the different B points you will use to determine a, as you did in Prelab Question 6. r Measure the height of each B with respect to the lab bench top and record it in the data table in your notebook. [6 pts] 20 r Your data table should now contain columns for the height at point A, the height at each point B, and the velocity of the ball as it passes each point B. [6 pts] 21 c.) Calculating a in Excel r Open Excel and copy your data into the worksheet. Use Excel to calculate vb 2. Copy the results into a new column in your notebook data table. r Plot your data in such a way that you obtain a straight line, the slope of which is related to a. Don t forget to label the axes (include units) and title your graph. (Hint: look at Question 6c.) r Fit a linear Trendline to the data and print the plot and Trendline equation. [9 pts] 22 r From the slope of the Trendline and your answer to Prelab Question 6d, determine a. Show all your work. [7 pts] Frictional Losses a.) Computer Setup r Quit the LoggerPro program. r Disconnect the first sensor from port 2 on the ULI box and connect the other sensor to the same port. r Click on the Lab05Centripetal icon and a velocity versus distance graph should appear as before, but this time the data will be coming from the other sensor. b.) Taking the Frictional Data r Sketch this experimental setup. [1 pt] 24 r Release the ball from point A and, as it enters the loop, click on Collect. r Data will appear on the computer screen once the ball has completed the loop and starts up the final incline. Click on Stop after the ball has reached its highest point and begins to descend down the track again. r Print this curve and attach it to your notebook. [3 pts] 25 If your data looks weird in any way, retake it until you get a nice smooth curve. r What is the speed of the ball when it has reached its final maximum height? (Your answer must be in a full sentence to receive any credit.) [3 pts] 26 r Using LoggerPro s X= icon, determine the distance between the motion sensor and the ball, just as the ball begins to roll back down the track. Record this distance in a new data table in your notebook. Title the data table appropriately. [4 pts] 27
6 Lab #5 Centripetal Motion/Conservation of Energy page 6 r Measure this stopping distance on the track and mark it with a piece of tape. r Measure the vertical distance between this stopping point and the top of the lab bench. Record this height (he) in your data table. [2 pts] 28 r Repeat these measurements two more times for two different A starting positions, recording each ha. Make sure you choose heights that are within 3 cm of each other (stay less than 80 cm). [20 pts] 29 r Calculate hloss,total = ha he and record them in your data table. [8 pts] 30 r Find the average value of hloss,total. [5 pts] 31 r The frictional energy loss in the track from A to D is roughly half the loss from A to E. Thus you will need to start the ball hloss, total/2 higher than the minimum height without friction, for the ball to make it all the way around the loop. You will determine this minimum height in the next section. r Calculate the average hloss,total/2, using the average hloss,total. Circle this number (label it and include units) in your notebook. [4 pts] Minimum Height Determination for Ball to go all the way around the Loop a.) Minimum Height with No Frictional Losses r Measure the loop diameter to determine its radius R. Show all work. [7 pts] 33 r Assuming (for now) there are no frictional losses, use your answer to Prelab Question 4b and the a you experimentally found in Part 1c, to determine the minimum height (hmin,no friction) needed for the ball to barely go around the loop. Don t forget to add in the height of the stand on which the apparatus is held. Show all your work; label and circle your answer. [6 pts] 34 b.) Minimum Height taking Frictional Losses into account r Now add to this hmin,no friction the extra height you found in Part 2b above (because of friction). This is the height at which you will need to release the ball, with friction present, for the ball to stay on the track all the way around the loop. [4 pts] 35 c.) Checking your Results r Try the ball from this height. Raise or lower the height as necessary until the ball barely goes all the way around on the track. Repeat this measurement 2 more times and measure the height of these points. Record the different heights in an appropriately labeled data table in your notebook. [6 pts] 36 r Find the mean of these new ha s. Explain how you did this; label your answer and circle it. [6 pts] 37 r Find the 99% confidence level of the standard deviation of the mean. Explain how you did this; label your answer and circle it. [6 pts] 38 r Calculate the % difference between your average measured value and your calculated value. How does your experimental value compare to your calculated value? [3 pts] 39
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