1 Module 2-3 Scaling Galileos Solar System 3 - Times Now we need to look at how long it takes for planets and moons to complete their orbits and how fast they are going. Quantitative Concepts and Skills Scale Direct proportions Unit conversions Periods and cycles Linear and angular velocity 2 PREVIEW The alumni do not want just a static model of the solar system. Rather their donation is for a working scale model -- where the globes (including the sand-sized ones) can be seen to move around in their various orbits. They like this idea because students would then get a feel for the passage of time in the solar system (this module) and how interplanetary distances vary with time (next module). We will scale the times so that Galileos outermost planet makes a complete round trip in 1 day. That 1 day compares to 29 years for the real Saturn to orbit the sun. So with that scaling factor what are the periods of the other planets and moons in Galileos system What will the speeds be4 scales the times.s 5 and 6 add the linear velocities ands 7 and 8 introduce and calculate angular velocities. s 9 and 10 calculate periods and velocities of the five moons in Galileos system.s 11-13 compare cycles of the moons to those of their planets. Your cell equations will need to include factors to convert time intervals from one unit to another. Among the conversions you might need to use (and remember) are 1 yr 365.25 days 1 day 24 hr 1 hr 60 min 1 min 60 sec 3 Retrieve this data sheet again. This time we will be using Columns F and G. 4 Scaling the time 1. Calculate E16 from C and D and G16 from E16. 2. Calculate E17 from E14 and F14. Then calculate F9F13 and G17 from E17. Columns C and D are from Modules 2-1 and 2-2. Column E is from data. We are scaling the model so that Saturn tracks around the campus in one day. 5 Scaling the time 2 Now work out the velocities. For the prototype planets calculate H from C and E and I from H. For the model planets calculate J from D and G and K from J. You will need to use the formula for the circumference of a circle. As a check you can also calculate K from D and F 6 Scaling the time 3 Side Exercise What is the scale for the velocity Calculate your answer in two different ways. In the Copernican system outer planets move more slowly than inner planets. This was a major point in Copernicuss argument against the Aristotelian system. 7 The velocities of6 are linear velocities the traveled distance divided by the elapsed time. Units for linear velocities are km/yr ft/day mi/hr among others. Angular velocity (this) is the angle swept out per unit time. Angular velocity is stated in units such as cycles/yr radians/hr degrees/min among others. Angular velocities For the prototype planets calculate E from C and F and G from E. For the model planets calculate H from D and I and J from H. You will need to use the number of radians per cycle and the number of degrees per cycle. Columns C and D are from previous. 8 Angular velocities 2 Side exercise What is the scale for the angular velocities How does this compare to the scale for the times 9 So far you have calculated the velocities of the planets around the sun (assumed stationary). You can easily calculate the velocity of the moons around their planets (assumed stationary). Adding the moons Columns C and E are from your data sheet Rows 11 and 14 are from previous spreadsheets but calculate them again to the third significant digit using the scales in E4 and E5. Then go on and complete the velocities for the moons. 10 Adding the moons 2 Notice how fast those moons of Jupiter are going In the scale model for example it takes a little less than a minute for Io to make four cycles around Jupiter In the same time Jupiter moves a bit more than four meters. 11 Moon cycles and planet cycles How many times does a moon go around its planet in the time that it takes for the planet to go around the sun Calculate D again from C and the scale in E4. Then calculate E from row data. 12 Moon cycles and planet cycles 2 Now calculate the moon-to-planet ratio of angular velocities. Calculate G from C and H from D (or alternatively H from C and E4). Then calculate I from the row data in G or H. 13 Moon cycles and planet cycles 3 Side exercise Explain why Columns E and I contain the same results. 14 End of Module Assignments
Do the three side exercises..
Add Uranus Neptune and Pluto to the spreadsheets ofs 6 and 8.
Add Uranus Neptune Pluto and the moons of Saturn Uranus Neptune and Pluto to the spreadsheets ofs 10 and 12.
The linear velocities of the moons you calculated in9 were relative to the planets that they circle. The velocities of the moons relative to the ground in the model would be something different. How would you calculate them
According to Cell E11 of12 our moon makes 13.38 cycles in the time that it takes our planet to circle the sun. Does this mean that we have 13.38 full moons in a year The answer is no. Look into the difference between sidereal and synodic periods. The periods that are relevant to the model are sidereal periods. Why
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