Kepler's laws, momentum and angular momentum In this lab you will study elliptic orbits and show that the momentum of an orbiting planet is not conserved, while the angular momentum is. Learning...

can i also get a picture of the online experiment and


Kepler's laws, momentum and angular momentum In this lab you will study elliptic orbits and show that the momentum of an orbiting planet is not conserved, while the angular momentum is. Learning outcomes: 1. Understand general elliptic orbits and how they are governed by Kepler’s laws. 2. Understand the applications and differences of momentum and angular momentum. Simulation Instead of using a physical apparatus, this lab uses the “Gravity and Orbits” open-source online interactive simulation provided by PhET. The simulation can be accessed at: https://phet.colorado.edu/en/simulation/gravity-and-orbits The simulation can be downloaded and run later without an internet connection. Part 1: Elliptical orbits Ellipse: An ellipse is a generalization of a circle, which has two diameters. The longer one is called the major axis and we will denote it as ?. The shorter one is called the minor axis and we will denote it as ?. The eccentricity characterizes how far away from a circle the ellipse is and is defined by ? = √1 − ?2 ?2 ? = 0: circle ? = 1: straight line The planets do not follow exactly circular orbits but, are slightly elliptical. Mercury has the greatest orbital eccentricity with ? = 0.2056 Start the simulation and select “To Scale”. Set the Star Mass to two times the Sun’s mass. Select “Path” to show the path of the planet and run the simulation. Question 1: Is the orbit of the planet circular or elliptical? Using the measuring tape, measure the minor axis and major axis of the orbit. What is the eccentricity of the orbit? Kepler’s laws: 1. The orbit of each planet around the Sun is an ellipse (with the Sun at one focus). 2. Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see the shaded figure to the right). 3. For two different orbits, the periods and major axis are related to each other by ?1 2 ?2 2 = ?1 3 ?2 3 Question 2: The Earth’s major axis is 300 × 106 km (twice its orbital radius) and its period is 365 days. Mercury’s major axis is 0.116 × 106km. Using Kepler’s third law, determine the orbital period of Mercury. We will use the simulation to check Kepler’s third law. The data for Earth’s standard orbit is recorded below in table 1. You will generate elliptical orbits by changing the starting position of the planet. You will then compare the elliptical orbits to Earth’s standard orbit. Reset the simulation and move the ? ? https://phet.colorado.edu/en/simulation/gravity-and-orbits planet closer to the Sun so that it is 75000 thousand miles away from the star. You can use the slider in the upper left to zoom in and out for easier measurements. Make sure the path and grid are checked and run the simulation. Let the planet make one full rotation and then pause the simulation. To help get exactly one rotation, set the speed to slow motion right before the planet finishes its orbit. • Record the time as the period ? in table 1 below. • Measure the largest diameter and record the result as ? in table 1 below. • Measure the smallest diameter and record the result as ? in table 1 below. • Compute the ratio of the square of the periods compared to Earth’s standard orbit and record the results as ?1 2 ?0 2. • Compute the ratio of the cube of the major axis compared Earth’s standard orbit and record the results as ?1 3 ?0 3. • Compute the eccentricity: ? = √1 − ?2 ?2 . Reset the simulation and repeat for the other two initial distance in table 1 below. Table 1: elliptic orbits and Kepler’s 3rd law Trial Initial distance (thousand miles) ? (thousand miles) ? (thousand miles) ? (days) ?? 2 ?0 2 ?? 3 ?0 3 ? 0 91500 186000 185950 365 0.02 1 2 3 Question 3: Does your data from table 1 confirm Kepler’s third law? Explain your reasoning. Question 4: Does the eccentricity of the orbit increase or decrease as you shorten the initial distance of the planet from the star? Why would this be the case? Part 2: Conservation of momentum and angular momentum Momentum: The momentum of an object is determined by the object’s velocity and mass as follows. ? = ? �⃗� Angular momentum: The angular momentum of an object is defined in terms of momentum similarly to how torque is defined in terms of force. If ? is the vector which points from the axis of rotation to the center of mass of the object and ? is the momentum of the object, then the angular momentum is given by ? = ? ? sin(?) where ? is the angle between ? and ?. In this simulation, all the measurements are at an angle of ? = 90°. ? ? ? Conservation of momentum: The momentum of an object or system is conserved whenever the total external force on the object or system is zero. �⃗�ext = 0 → ∆? = 0 Conservation of angular momentum: The angular momentum of an object or system is conserved whenever the total external torque on the object or system is zero. ?ext = 0 → ∆�⃗⃗� = 0 Question 5: Is the momentum of a planet conserved as it orbits a star? Explain your answer using force. Question 6: Is the angular momentum of a planet conserved as it orbits a star? Explain your answer using torque. Our goal will be to determine if the momentum and angular momentum of the planet is conserved as it orbits the star. Restart the simulation and move the plant so that it is 50000 thousand miles from the star. The first goal will be to measure the speed of the planet at four different locations along the orbit, as well as the distance from the planet to the star. Measuring the velocity and orbital distance: • Pause the simulation slightly before the place you want to measure the velocity. • Reset the path by clicking the path box off and on. • Reset the timer by clicking clear under the timer. • Un-pause the simulation and let it run a short amount of time. The grey path should be short enough so that it is roughly a straight line, as shown in the figure to the right. • Use the measuring tape to measure the length of the grey path, which is the distance the planet traveled. Record the measurement as ∆? in table 2 below. • Record the time as ∆? in table 2 below. • Use the measuring tape to measure the distance between the star and the center of the planet’s path. Record the measurement as ? in table 2 below. Repeat for four more points along the orbit to fill out the table below. Try to sample from various locations along the planet’s orbit. After you have your measurements, convert them to SI units and record your results in table 2. ? ∆? ∆? = Table 2: Planet speed and orbital distance ∆? (thousand miles) ∆? (days) ? (thousand miles) ∆? (m) ∆? (?) ? (?) Using your above data, compute the speed, momentum and angular momentum of the planet. Note the angle between the velocity �⃗� and the vector ? is always 90°. We will assume the planet is Earth and use Earth’s mass for the mass of the planet. Table 3: Momentum and angular momentum ? (kg) ? (m/s) ? (kg m/s) ? (kg m2/s) 5.972 × 1024 Question 7: Is the momentum of the planet conserved? Explain your answer using your data. Question 8: Is the angular momentum of the planet conserved? Explain your answer using your data. Question 9: In science fiction movies, space pilots will often refer to the “slingshot” effect as a way to generate large speeds. Explain how the slingshot effect works in terms of conservation of angular momentum. Lab report: Kepler's laws, momentum and angular momentum Name:___________________Date: ____________ Course: __________ Question 1: Is the orbit of the planet circular or elliptical? Using the measuring tape, measure the minor axis and major axis of the orbit. What is the eccentricity of the orbit? Question 2: The Earth’s major axis is (twice its orbital radius) and its period is . Mercury’s major axis is . Using Kepler’s third law, determine the orbital period of Mercury. Table 1: elliptic orbits and Kepler’s 3rd law Trial Initial distance 0 1 2 3 Question 3: Does your data from table 1 confirm Kepler’s third law? Explain your reasoning. Question 4: Does the eccentricity of the orbit increase or decrease as you shorten the initial distance of the planet from the star? Why would this be the case? Question 5: Is the momentum of a planet conserved as it orbits a star? Explain your answer using force. Question 6: Is the angular momentum of a planet conserved as it orbits a star? Explain your answer using torque. Table 2: Planet speed and orbital distance Table 3: Momentum and angular momentum Question 7: Is the momentum of the planet conserved? Explain your answer using your data. Question 8: Is the angular momentum of the planet conserved? Explain your answer using your data