Date: Problem Set # 1. Unit conversion is an important part of any scientific research. Please view the following link: c = (gh) α (3)

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1 Problem Set # 1 Instructions: Numerical answers require units and appropriate numbers of significant digits. Remember to show your work! Review Problems: R-1. (3 points) For the function f(x) = 4x 3 6x + 2 (1) (a) Find the indefinite integral, F(x), of this function. (b) Evaluate the constant of integration if F(x = 2) = 23. (c) Check your answer by differentiating F(x) to show that df(x) dx = f(x) (2) R-2. (4 points) Unit conversion: Unit conversion is an important part of any scientific research. Please view the following link: Here, we are going to work on a simple example. Speed of tsunamis, c, depends only on the ocean depth and is given by the following formula in the high seas. c = (gh) α (3) where g is the acceleration due to gravity and h is the ocean depth at any point. EARTH Fall 2017

2 (a) Using the units (dimensions), find the exponent α. (b) Assuming that the typical ocean depth is 4000 m, calculate the typical speed of tsunami waves in m/s. (c) Calculate the tsunami speed in km/h. (d) Calculate the tsunami speed in mi/h. (e) How do these numbers compare with the speed of a jet airliner? 1. (4 points) Consider the equations for the gravitational force and the electrostatic force: F g = Gm 1m 2 F r 2 E = kq 1q 2 r 2 (a) Compare the two equations. In what ways do they differ? (b) A large ship (mass = kg) is out at sea when a small fish (mass = 2 kg) swims 1 meters below it. How much force does the fish exert on the ship? How much force does the ship exert on the fish? Should the fish be worried about getting pulled to the ship? Why or why not? 2. (5 points) To determine the size of the moon, use the fact that it appears to be the same size as a dime held 154 cm from the observer (this is called having the same angular diameters). If the moon is km away, what is the radius of the moon? How does the moon compare in volume to the earth? Figure 1: Dime and the Moon. EARTH Fall 2017

3 3. (5 points) The acceleration of gravity on the lunar surface is about 1 that on the earth s 6 surface. Given the moon s radius from problem 2, find its mass and average mean density. 4. (3 points) Your lazy roommate thinks that instead of actually working on problem 3, you can instead say that if the acceleration of gravity on the moon was 1 that on earth, its mean 6 density is 1 that of earth. Why isn t this correct? 6 5. (3 points) A model often used for the moon is that it is made of (green) cheese. Test this model by comparing its density to a block of Monterey Jack with dimensions 4 cm 6 cm 6 cm and a mass of 143 grams. 6. (4 points) Show how gravity at the Moon s surface would change if its: (a) Radius is kept constant, but density is doubled; (b) Radius is doubled, but density is kept constant. 7. (6 points) We saw in class that a body in orbit has an orbital period which depends on its distance from the center of the planet being orbited. Some satellites are put in synchronous orbit that is, with a period equal to one planetary day so that they stay above the same point on the surface as the planet rotates. For the planet Mars, how large is the radius of this orbit? How high above Mars s surface is the satellite? 8. (4 points) Moment of inertia factors I/MR 2 (and other information) for the various planets can be found on the website or on the class website by clicking on the planetary fact sheet, which is found underneath TOPIC 1: Size, Mass, & Density of the Earth. Find the values for the Sun, Venus, Mars, the moon, Earth, and Jupiter. Put these values in order from largest to smallest and explain what they tell about the density distribution (the way that the density changes from the surface to the center of an object). EARTH Fall 2017

4 9. (8 points) The formulas for calculating the mass and moment of inertia of a planet are: M = 4π a 0 ρ(r)r 2 dr I = 8 a 3 π ρ(r)r 4 dr 0 Given these formulas, show that a two-shell planet with mantle density ρ m, core density ρ c, radius a, and core radius r c, has a mass and moment of inertia: M = 4 3 π[ρ ma 3 + (ρ c ρ m )r 3 c] I = 8 15 π[ρ ma 5 + (ρ c ρ m )r 5 c] C-1. 1 (8 points) Using the results of problem 9: (a) Write a spreadsheet or program that takes as inputs a, ρ m, ρ c, and r c and computes r c /a, I, M, and I/Ma 2 (Hint: In Excel the symbol ˆ is used to raise a quantity to a power). (b) Test this by setting ρ m = ρ c. What is I/Ma 2 and why? (c) Test this by setting ρ c = 0 and r c = 0.99a. What is I/Ma 2 and why? (d) Derive a plausible two-layer model for Mercury, assuming I/Ma 2 = In addition to your answers for parts (a) to (d), please hand in a printout or screenshot of your code or excel spreadsheet. C-2. (10 points) Use the uploaded coordinates from the entire class (on the online spreadsheet) in Lab#0 and follow these steps to make a histogram of latitudes and longitudes of the measurements. Note: Watch the YouTube video at to see how you can make histograms in Excel. (a) On a new spreadsheet on your computer, copy and paste all the longitudes in a single column and all the latitudes in another one as shown in Fig Problems numbered as C-# are computer problems. EARTH Fall 2017

Figure 2: Copy and paste the values for longitude and latitude into a spreadsheet. (b) In a third column, list the 0.00003 bins between 87.676950 87.676610 in longitude and between 42.057750 42.

5 Figure 2: Copy and paste the values for longitude and latitude into a spreadsheet. (b) In a third column, list the bins between in longitude and between in latitude. (c) Use Excel s Data Analysis tool under the Data tab to make a histogram. Note: If the Data Analysis tool is not available on your computer, you can enable it at Tools>Excel Add-ins>Analysis ToolPak. (d) Turn in a printout of your histogram attached to this handout. C-3. (5 points) Use Google Earth to plot the measurements on a map: (a) Save the only two columns in the spreadsheet you made in C-2 in the CSV (comma sparated) format which is going to be a text file. (b) Open Google Earth, and click on File>Import; from the dialog box that opens, select your CSV file. EARTH Fall 2017

6 (c) In the new dialog box, make sure the Comma option is checked and also the Field Type is set to Delimited. Click Next. (d) In the drop-down menue, set the Latitude Field to the the one that begins with and the Longitude Field to the one that begins with Click Next. (e) Make sure the field types are set to floating point. Click Finish. (f) You are asked if you want to apply a template to the features you ingested. Choose Yes. (g) Check Create new template. Click OK. (h) You can edit the appearance of your points on the map by setting the attributes (colors & icon) here. When your are done, click OK. save your template if you want. (i) On the left side pane, double click on your file name, and make sure to check the boxes next to your points to make them appear on the map. (j) Zoom into the map until you get a close view of the points, and then save the view as an image. (k) Attach the image to this handout before you turn it in. C-4. (5 points) Usually, geospatial measurements are considered to be accurate(=exact) in geodesy. For istance if you double click/tap on a fixed point (say the corner of the planter which you measured in Lab#0) in Goole Earth or Google Map, you get coordinates that are fixed. But the values you got for your location s coordinates (the blue dot in Google Map) vary every time you make a measurement. Try to quantify how the set of measurements by the entire class compares to the reference value, i.e W, N. How accurate/precise are the measurements? (Hint: think in terms of resolution) EARTH Fall 2017

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