Circular Motion Gravitation Labs

Centripetal Force Exploring Uniform Circular Motion An object that moves in a circle at constant speed, v, is said to experience uniform circular motion (UCM). The magnitude of the velocity remains constant, but the direction of the velocity is continuously changing, as shown in Figure 1. Figure ] Notice that the velocity vectors are not pointing in the same direction. The velocity vector is always tangent to the circular path. This is why, should the string break, the object flies off in a straight line. Figure 2 An object revolving in a circle is continuously accelerating, even when the speed remains constant. The acceleration vector is always pointing toward the center of the circular path and it is always perpendicular to the velocity vector, as shown in Figure 2. Centripetal (center-seeking) acceleration, a., is defined as: Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

10. a. Mass the rubber stopper and record its mass in kilograms in Data Table 1 on your student answer page. Attach one end of the cord securely to the rubber stopper. b. Pass the other end of the cord through the glass or plastic tube. ¢. Bend a large paperclip into a hook and attach it to the free end of the cord. This hook needs to support several large washers. . The other partner should use a balance to mass your group’s washers + hook. Keep adding washers until the mass exceeds 100 grams. Record the mass of your stack of washers and hook in kilograms in Data Table 1 on your student answer page. Load the stack of washers onto the hook. If the stack is too large to hang securely from the hook, you can secure the stack together with a piece of string and hang the string from the hook. If you choose this method, re-weigh your hook as well as the stack of washers with the string attached. Adjust the cord so that there is about 0.75 m of cord between the top of the tube and the stopper. The partner that is going to operate the apparatus should support the stack of washers in one hand and hold the tube with the other. Begin whirling the stopper by moving the tube in a circular motion. Slowly release the stack of washers and adjust the speed of the stopper so that the stack of washers remains relatively stationary, meaning the stack is no longer climbing or falling. Important: Keep the stopper whirling in an elevated, horizontal circle as pictured in Figure 4. The tube should be held high enough in the air so that the spinning stopper does not threaten any part of the spinner’s head. Make several trial runs to master the technique. Be mindful of your space so that you do not accidentally hit any other students with your spinning stopper. When you have learned how to keep the velocity of the stopper and the position of the washers relatively constant, have your partner measure the time it takes for 20 revolutions by counting down the revolutions “5,4,3,2,1, go! 1,2,3, .... ” Record this time in Data Table 1 on your student answer page. The partner with the apparatus should stop the whirling stopper by placing his or her finger at the fop of the tube so as to capture the length of the radius of your circle. Use caution when performing this task so as not to be hit by the stopper. Once the stopper comes to rest, keep your finger in place so that the length of the cord will not change. Measure the radius, r, from the center of the tube to the center of the stopper. Use the appropriate number of significant digits and record the value in Data Table 1 on your student answer page. Repeat the procedure for two additional trials keeping the same stopper and the same stack of washers, but varying the radius. Keep the radius between 0.50 m and 0.90 m. Record all data to the appropriate number of significant digits in your data table. Copyright © 2012 Laying the Foundation®, Inc., Dalfas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

ANALYSIS 1. Calculate the force of weight, Fiy, of the hanging mass and enter it in Table 2 as centripetal force, F.. F, =mg=F, and g=9.8fl2 s 2. Calculate the period of revolution by dividing the total time by the number of revolutions and enter it in Table 2. time(s) - 20 revolutions 3. Calculate the circumference of revolution from the radius and enter it inTable 2. circumference = zd = 27zr 4. Use the circumference and period to find the velocity and enter it in Table 2. circumference V = - T 11010 (PR OF.N D @) B) B7.§ 4 (O AN Trial Centripetal Force, F Period, T Circumference Velocity N) (s) (m) (m/s) 1 2 3 4 5 6 7 8 9 Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

9. As you whirled the mass in a circular path, you did your best to keep it swinging in a horizontal circle. Is it possible to keep the mass whirling in a perfectly horizontal circle (parallel to the floor)? Explain your answer and draw and label the forces acting on the mass from a side view using the diagram below. i B 10. A student tries to swing a stopper of mass 40 grams in a horizontal circle, but finds that the stopper “droops” down a little due to the gravitational force acting on the stopper, as shown in the figure below. The string has a length L = 0.50 m, and is swung so that it remains an angle of 8 = 60° from the vertical. Sy O el e i e a. On the diagram above, draw and label all of the forces acting on the stopper as it travels around the circle. (Remember: centripetal force is not an “extra” force that acts on the mass, but is the sum of the forces which point toward the center of the horizontal circle.) b. Can we say that the tension in the string is the centripetal force? Explain your answer. c. Show the horizontal and vertical components of the tension force on the diagram above. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

Physics SITF LAYING THE FOUNDATION Building Academic Excellence The Pendulum Swings Happiness Is A Straight Line In this activity, you will determine the relationship between the period 7 of a pendulum and its mass m, amplitude 4, and length L. Your teacher will instruct you on how to construct your pendulum. You will make and record measurements throughout this activity, and you will create your own graphs to display your data. Purpose You will investigate the relationship between the mass, amplitude, and length of a pendulum and its period. From the data collected, you will also apply your graphing skills either manually or by using computer graphing software to generate graphs and find relationships. You will also linearize your data if necessary. Materials Each lab group will need the following: meter stick string protractor, with hole tape, masking scissors 3 washers, 2 in. stopwatch Copyright © 2012 Laying the Foundation®, inc., Dallas, Texas. All rights reserved. Visit us online gt www.ltftraining.org. 1

Student Activity — The Pendulum Swings Conclusion Questions 1. What are the variables in this experiment? 2. What data should be recorded for your pendulum? 3. What steps did you take to maximize accuracy and precision, and minimize your systemic error while collecting your data? 4. What effect does varying the mass of the pendulum have on its period? Justify your answer. 5. What effect does varying the amplitude of the swing of the pendulum have on its period? Justify your answer. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.itftraining.org. 4

Simple Harmonic Motion Investigating a Mass Oscillating on a Spring A spring that is hanging vertically from a support with no mass at the end of the spring has a length L (called its rest length). When a mass is added to the spring, its length increases by x. The equilibrium position of the mass is now a distance L + x from the spring’s support. The spring exerts a restoring force, ' = -kx, where x is the distance the spring is displaced from equilibrium and k is the force constant of the spring (also called the spring constant). The negative sign indicates that the force exerted by the spring on the mass is directed opposite to the direction of the displacement of the mass. The restoring force causes the mass to oscillate up and down, and is always directed toward the equilibrium position. The period of oscillation depends on the mass and the spring constant. PURPOSE In this lab you will determine the spring constant for a spring, predict the period of oscillation for several different masses using the measured spring constant, and find their relative error. A graph of position vs. time and velocity vs. time for a mass in simple harmonic motion will be constructed and analyzed for the energy aspects of the motion. MATERIALS Each lab group will need the following: calculator, TI® graphing clamp, pendulum computer electronic file LabQuest® meter stick ring stand sensor, motion detector spring weights, hanging sets Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.itffraining.org.

PROCEDURE PART I: FINDING THE SPRING CONSTANT & 1. Attach a clamp to a ring stand so that your spring will hang over the edge of the table. I 1~ Z v Figure 1 2. Hang a mass hanger on the spring so that it will slightly stretch your spring and hang at rest. With your meter stick, measure the distance from the bottom of the hanger to the floor where the motion detector will eventually be positioned. Make all successive measurements from this same position, which will serve as the equilibrium position. This will be considered your unstretched length. 3. Measure the amount of stretch for 5 different masses added to the spring. The amount of stretch is the difference between the unstretched length and the stretched length. Record your measurements in Data Table 1 on your student answer page. PART II: ADDITIONAL METHOD FOR FINDING THE SPRING CONSTANT K 1. Attach a motion detector to your computer and start the appropriate software. 2. Place the motion detector on the floor directly beneath the spring. Make sure that the mass does not come closer than the measurement limit for your motion detector, usually 15 cm for Vernier motion detectors or PASCO Motion Detector II’s. When Logger Pro is opened, if it does not automatically identify your motion detector, then drag the motion detector icon to Digital Channel 1. A graph of position vs. time and a second graph of velocity vs. time should open. "W I~ Figure 2 Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

3. Using each of the 5 masses from Part I, set the mass into smooth oscillations, with amplitudes of about 10 cm (depending on the original length of your spring). Start the motion detector data collection. Gather data for about 3 seconds or until you have more than 10 complete oscillations with each of the 5 masses. 4. Using the analysis tool on Logger Pro or your software, find the time for ten complete oscillations of your spring. Record this, as well as the mass on the spring, in Table 2 on your student answer page. PART III: ENERGY IN SIMPLE HARMONIC MOTION 1. With the motion detector placed on the floor as in Part II, zero the motion detector when the mass is motionless. To zero the motion detector, click on the LabPro icon on the tool bar. Double click on the motion sensor icon, and then select “zero”. This is your equilibrium position. This will have to be repeated with each mass. 2. Set the mass into smooth oscillations with amplitude of about 10 cm (depending on the original length of your spring). Collect data for about three seconds. 3. Using the analysis tool, determine both the position and velocity for the mass at its maximum amplitude and for the next position of zero amplitude. Record these values in Table 3 on your student answer page. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

Simple Harmonic Motion Investigating a Mass Oscillating on a Spring DATA AND OBSERVATIONS PART I: DETERMINING THE SPRING CONSTANT k Data Table 1 Run # | Mass on Spring | Force [Position 1 with only| Position 2 with Amount of (kg) N) hanger added mass stretch (m) (m) (m) 1 2 3 4 5 Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

ANALYSIS PART I: DETERMINING THE SPRING CONSTANT K 1. On the mass below, draw and label the forces acting on the mass when it is hanging at rest on the end of the spring. I 2. Using either graph paper or graphing software, construct a graph of Force (N) vs. Amount of Stretch (m). Find the slope of this graph. a. What are the units for the slope? b. The equation relating the magnitude of the force and the stretch is F = kx . How does this equation relate to the slope of your graph? Force vs Stretch - Linear Fit For: Data Set:Force y= mx+b /fl m(Slope): 2.85 Nim 1.0 b(Y-Intercept): 0.117 N / Correlation:0.999 [l (O] S p i <4 o [F 0.5 /@/ 1 1 I I T L) T I | L) L) T 1 0.10 0.20 0.30 Stretch ¢ (m) (0.247568, 0.662008) Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.itftraining.org.

Data Set Streteh (%) Force (m) ™ | 1 0.04400 0.29500 | & PEIEen 0.11500 0.44100 3 0.18500 0.63700 AN 0.25900 0.89300 5 0.32500 1,03000 . 4 || PART IlI: ADDITIONAL METHOD FOR FINDING THE SPRING CONSTANT K 1. The period of oscillation for your mass spring system can be calculated using the following equation. Rearrange this equation to solve for the spring constant k, and show your work below. For each of your five Runs, calculate a value for the spring constant and then find the average of these values. Place your calculations in Table 2. T=27r\/E k Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. Al rights reserved. Visit us online at www.ltftraining.org.

i 1) [ Mass on Spring Time for 10 Measured Period | Calculated value for Run # (kg) oscillations (s) k (s) (N/m) 1 2 3 4 5 Avg. n/a n/a n/a 2. Compare the average value for & that you calculated in Part II to the value you found for & in Part I. Find a relative error for the two values. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

PART III: ENERGY IN SIMPLE HARMONIC MOTION 1. For this oscillating system, the energy of the system should remain constant for short periods of time involving one or two oscillations. Over time friction will convert some of the energy to heat, but for a single oscillation, the elastic potential energy Us of the system at maximum amplitude should be relatively close to the kinetic energy KE when the system returns to the equilibrium position. Calculate these values and compare them by finding a relative error. U, = L T2 1 KE =—mv 2 : , — KE| Relative Error = x100 Table 3 Mass on | Amplitude Velocity at Calculated [Calculated] Relative Run # Spring | of stretch [equilibrium position EPE KE Error (kg) (m) (m/s) Q) @ (“e) | |WIN|— 2. The errors in this part of the lab may be relatively large. Write a statement analyzing the errors in this lab and suggest ways that the errors might be reduced. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

CONCLUSION QUESTIONS 1. A person who weighs 670 N steps onto a spring scale in the bathroom, and the spring compresses by 0.79 cm. (a) What is the spring constant? Be sure to specify your units. (b) What is the weight of another person who compresses the spring by 0.34 cm? 2. What is the shape of the plot of data for the oscillating spring-mass system on your graph? Describe the shape physically. 3. In terms of the changes in force, displacement, acceleration, and energy, describe one full oscillation of a mass on a spring. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.

4. When a 2.8-kg object is suspended from a spring, the length increases by 0.018 m. If the frequency of vibration is f= 3.0 Hz, how much mass is attached to this spring? 5. What is the maximum speed of the mass in question 4? Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltffraining.org.

Physics SITF LAYING THE FOUNDATION Building Academic Excellence Waves in a Spring Observing the Characteristics of Waves Purpose In this activity, you will observe and investigate important wave properties. What you observe about waves in a spring can be applied to other types of waves including sound waves, water waves, and light waves. Materials Each lab group will need the following: meter stick spring (slinky) stopwatch spring, 6 ft (snakey) SAFETY ALERT! » Coiled springs are social creatures. Avoid releasing either end of the stretched spring. The untangling process can be quite difficult. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 1

Student Activity — Waves in a Spring Procedure Answer the questions as you perform this activity. You and your lab partner should hold opposite ends of a spring and stretch it out on the floor to an appropriate length. Experiment to determine the best stretch distance for your spring but be careful not to exceed the elastic limit of the material. Data and Observations Longitudinal and Transverse Waves 1. Pinch a clump of coils together with your free hand and release the clump of coils. Do not let go of the end of the spring. Observe the pulse that travels back and forth through the spring. Why is it called a longitudinal pulse? Sketch the wave pulse in the space provided. 2. Give one end of the spring a few vigorous sideways (fransverse) shakes. How is this wave different from the longitudinal wave? Sketch the wave pulse in the space provided. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 2

Student Activity — Waves in a Spring Data and Observations (continued) 3. The stretched spring is the medium through which the pulse travels. Send a short transverse pulse down the spring. Observe the shape of the pulse as it moves along the spring. a. How does the shape change? Can you suggest a reason for this change? b. Upon what does the initial amplitude of the pulse depend? c. Does the speed of the pulse appear to change with its shape? d. Generate single pulses of various (small, medium, and large) amplitudes. Does the pulse speed appear to depend on the size of the pulse? e. What could you do to produce wave pulses that travel faster in the spring? Can this be done by shaking the spring more vigorously? How about shaking the spring faster? Copyright © 2012 Laying the Foundation®, inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 3

Data and Observations (continued) Student Activity — Waves in a Spring 4. Measure the length of a stretched spring (do not change this length) and the travel time of a pulse generated at one end. Use a stopwatch to time how long a wave takes to travel up and down the spring. Calculate the speed of the traveling wave pulse for pulse sizes that are small, medium, and large. Record your results in Table 1. Is the speed of the wave pulse affected by the amplitude of the pulse? Use your data and d = vt to answer this question. Transverse Small Transverse Medium Transverse Large Longitudinal -- 5. Change the length of your stretched spring and determine the pulse speed as you did previously. Record your results in Table 2. Examine three different lengths for the stretched spring. Is the speed of the wave pulse affected by the length of the stretched spring (the tension)? Under different tensions, does the stretched spring represent the same or different media? Table 2. Spring Length Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 4

Student Activity — Waves in a Spring Data and Observations (continued) Interference 6. Have your lab partner generate a wave pulse toward you on the same side of the spring that you send a pulse toward your partner. This interaction between the two wave pulses is called interference. a. Describe the interference of the two wave pulses. How does the pulse amplitude during interference compare with the individual amplitudes before and after this interaction? b. Repeat the experiment but with the two pulses traveling on opposite sides of the spring. Compare the interference with that of the previous interaction. When the two pulses meet, does the displacement of the spring at that instant get larger or smaller? c. Did the two pulses pass through each other or bounce off each other? If you are not sure, have your partner send a transverse wave pulse down the spring at the same time you send a longitudinal wave pulse. What can you say about the interaction of these two wave pulses? Do the wave pulses bounce off each other or do they pass through each other? d. What conclusions can you draw about the displacement of the medium at a point where two pulses interfere? (This is called the principle of superposition.) Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.itftraining.org. 5

Student Activity — Waves in a Spring Data and Observations (continued) Reflection 7. With the far end of the spring held firmly in place by your lab partner (the “fixed end”), send a single pulse down one side of the spring. Observe the reflected pulse. a. Compare the amplitude of this pulse with that of the transmitted pulse just before reflection. What is the orientation of the reflected pulse relative to the transmitted pulse? b. Attach a light string about 2 m long to the far end of the spring, and maintain the tension on the spring by holding the end of this string. This is called a free-end termination for the spring. Send a pulse down one side of the spring as before and observe the pulse reflected from the “free” end. Compare the reflected pulse from the “free” end of the spring with the reflected pulse from the fixed end. Periodic and Standing Waves 8. By moving your hand steadily back and forth, you can produce a series of pulses called a periodic wave. The distance between any two adjacent crests or troughs on a periodic wave is called the wavelength. The rate at which you move your hand back and forth determines the frequency. Generate a periodic wave and take a “snapshot” of the wave. Sketch the picture on the axes provided in Figure 1. On your diagram, show the amplitude of the periodic wave. Sketch one wavelength and, using the total distance of the wave, determine the value for this wavelength. How does the wavelength depend on the frequency? distance displacement o Figure 1. Periodic wave Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltffraining.org. 6

Student Activity — Waves in a Spring Data and Observations (continued) 9. Have your partner generate a continuous periodic wave while you try and match it. You have produced a standing wave. While your partner holds the end of the spring steady, you will form transverse standing waves having first one, two, three, and then four loops. Find the frequency needed to produce each standing wave pattern. The frequency is the number of times your hand moves through a complete cycle (back and forth) every second. It will be easier to find the frequency if you time 10 cycles of your hand. After you find the frequency, you will be able to find the speed of the waves just created using v = f4. Table 3. Standing Waves 2 A=L 3 A=%x%xL 4 A=Y xL Record your results in Table 3, where L is the distance between both lab partners holding either end of the spring. a. What are the loops called? b. What are the places called where the spring does not appear to move? c. What is the general relationship between the number of loops and the frequency of the wave? Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 7

Student Activity — Waves in a Spring Analysis 1. How does the product of frequency and wavelength (fA) for the various standing wave patterns compare with one another? 2. How do these values compare with the speeds of the pulses calculated for the transverse wave? 3. How are the speed, frequency, and wavelength of a transverse wave related to one another? 4. Consider the power (number of joules of work per second) you are expending as you sweep your hand back and forth to create transverse standing waves. Does the power you expend to create waves depend on the frequency of the waves? 5. Does the power required depend on the amplitude of the wave? Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.itftraining.org. 8

Student Activity — Waves in a Spring Conclusion Questions 1. Suppose a flute player and a tuba player start playing at the same time, each from equally far away. Which instrument will you hear first? Why? 2. Which travels faster: red light or blue light? Blue light has a greater frequency. 3. For the waves shown in Figure 2, sketch the sum of the two waves. a ? d e b | I 0 c -1 -2 = = 4 3 2 1 0 -1 -2 Figure 2. Sum of two waves Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 9

Student Activity — Waves in a Spring Conclusion Questions (continued) 4. What happens to the displacement when two waves meet in phase (crest on crest)? 5. What happens to the displacement when two waves meet out of phase (crest on trough)? 6. What happens to the pulses after they pass through each other? 7. Explain how the reflection of a wave from a rigid barrier (a fixed end) and a less rigid barrier (a free end) affect the wave’s phase. 8. What happens to the speed, wavelength, and frequency of a wave when it reaches the boundary between two media? Copyright © 2012 Laying the Foundation®, inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 10

Student Activity — Waves in a Spring Conclusion Questions (continued) 9. A string is attached to a vibrating machine that has a frequency of 120 Hz, as shown in Figure 3. The other end of the string is passed over a pulley of negligible mass and friction, and is attached to a weight hanger that holds a mass m = 0.5 kg. Figure 3. Vibrating machine with standing wave a. Determine the tension in the string. b. The speed of the wave in the string is related to the tension by the equation where F_ is the tension in the string and 4 is the linear density of the string. If the linear density of this string is 0.05 kg/m, determine the speed of the wave in the string. c. Determine the wavelength of the wave in the string. d. Determine the length of the string from the point of attachment on the vibrating machine to the pulley. e. Would you need to increase or decrease the mass on the hanger to produce fewer loops? Explain. Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org. 11

Explor#learning ................................................................................................................ Student Exploration: Orbital Motion — Kepler’s Laws Vocabulary: astronomical unit, eccentricity, ellipse, force, gravity, Kepler's first law, Kepler's second law, Kepler's third law, orbit, orbital radius, period, vector, velocity Prior Knowledge Questions (Do these BEFORE using the Gizmo.) 1. The orbit of Halley’s Comet, shown at right, has an oval shape. In which part of its orbit do you think Halley’s Comet travels fastest? Slowest? Mark these points on the diagram at right. 2. How might a collision between Neptune and Halley’s Comet affect Neptune’s orbit? Gizmo Warm-up The path of each planet around the Sun is determined by two factors: its current velogity (speed and direction) and the force of gravity on the planet. You can manipulate both of these factors as you investigate planetary orbits in the Orbital Motion — Kepler's Laws Gizmo™. On the CONTROLS pane of the Gizmo, turn on Show trails and check that Show vectors is on. Click Play (). 1. What is the shape of the planet’s orbit? 2. Watch the orbit over time. Does the orbit ever change, or is it stable? 3. Click Reset ((©)). Drag the tip of the purple arrow to shorten it and reduce the planet’s initial velocity. Click Play. How does this affect the shape of the orbit?

Activity A: Shape of orbits Get the Gizmo ready: ¢ Click Reset. e Turn on Show grid. 7~ -1T123 Introduction: The velocity of a planet is represented by an arrow called a vector. The vector is described by two components: the i component represents east-west speed and the j component represents north-south speed. The unit of speed is kilometers per second (km/s). Question: How do we describe the shape of an orbit? 1. Sketch: The distance unit used her is the astronomical unit (AU), equal to the average Earth-Sun distance. Place the planet on the i axis at r = -3.00i AU. Move the velocity vector so that v = -8.0j km/s (|v| = 8.00 km/s). The resulting vectors should look like the vectors in the image at right. (Vectors do not have to be exact.) Click Play, and then click Pause ((CD) after one revolution. Sketch the resulting orbit on the grid. [ N w &~ o arta,=by+ b i = el LelbA-5C W 2. Identify: The shape of the orbit is an ellipse, a type of flattened circle. An ellipse has a center (C) and two points called foci (F1 and F,). If you picked any point on the ellipse, the sum of the distances to the foci is constant. For example, in the ellipse at right: Turn on Show foci and center. The center is represented by a red dot, and the foci are shown by two blue dots. What do you notice about the position of the Sun? 3. Experiment: Try several other combinations of initial position and velocity. A. What do you notice about the orbits? B. What do you notice about the position of the Sun? You have just demonstrated Keplér’s first1aw, one of three laws discovered by the German astronomer Johannes Kepler (1571-1630). Kepler's first law states that planets travel around the Sun in elliptical orbits with the Sun at one focus of the ellipse. (Activity A continued on next page) Wizmos

Activity A (continued from previous page) 4. Observe: Use the Gizmo to create an orbit that is nearly circular. Then create an orbit that is flattened. Observe the foci in each ellipse. A. What do you notice about the spacing of the foci when the ellipse is very round? B. What do you notice about the spacing of the foci when the ellipse is very flat? 5. Calculate: The eccentricity of an ellipse is a number that describes the flatness of the ellipse. Eccentricity is equal to the distance between foci divided by the total width of the ellipse. There are no units for eccentricity. Click Reset. Move the planet to r = —5.00i AU (does not have to be exact) and drag the velocity vector to set the velocity close to —8.0j km/s. Click Play, and then click Pause after one full revolution. A. What is the distance between the foci? B. What is the approximate width of the ellipse? C. What is the eccentricity of the ellipse? D. Click Reset, and change the initial velocity to —4.0j km/s. Click Play. What is the eccentricity of this ellipse? Distance between foci: Width: Eccentricity: 6. Draw conclusions: Think about the eccentricity and shape of each ellipse. A. What is the relationship between the eccentricity of an ellipse and its shape? B. What is the eccentricity of a circle? Explain. C. What is the eccentricity of a completely flat ellipse? Explain. The eccentricity of Earth’s orbit is 0.17. What can you infer about the shape of Earth’s orbit? @}izmos

Activity B: Get the Gizmo ready: \"[ o Velocity and area P ClcREse Tl 2__, 3 y e Turn off Show foci and center. Introduction: After establishing that planetary orbits were ellipses, Kepler next looked at the speed of a planet as it traveled around the Sun. Question: How does the velocity of a planet vary as it travels in its orbit? 1. Observe: Place the planet at —5.00i AU and set the velocity to -4.0j km/s (does not have to be exact). Turn off Show vectors. Click Play. Observe the speed of the planet. A. At what point does the planet move fastest? B. At what point does it move slowest? 2. Observe: Click Reset, and turn on Show vectors. Look at the green vector that represents the force of gravity pulling on the planet. Click Play. A. In which direction does the green vector always point? B. In which part of the orbit does the gravity vector point in almost the same direction as the velocity vector? C. In which part of the orbit does the gravity vector point in the opposite direction as the velocity vector? 3. Explain: Based on your observations of the gravity vector, why does the planet accelerate as it approaches the Sun and slow down as it moves away from the Sun? 4. Measure: Click Reset. Imagine a line connecting the planet to the Sun. As the planet moves around the Sun, the line will sweep out an area. Click Play, and then click Sweep area. Record the area below, and then press Sweep area four more times to complete the table. Trial 1 2 3 4 5 Area (km?) (Activity B continued on next page)