Physics 109 Homework #1

Due Thursday September 13, 2:00 pm.

Following Thales: recall Thales learned in Egypt how to measure heights by measuring a shadow, and how to find distances of inaccessible objects by measuring angles and drawing scale model triangles. I want you to try this on the Grounds. You will need a ruler and a protractor. I think you should work in groups of two or three, but each of you should write up the homework by yourself, using notes if you like.

1. (a) Go to the point across Emmet from the Bookstore Parking Garage, where you’ll see lamp posts like the one pictured below. Find the height of the lamp post by measuring the length of its shadow (pace it out) and comparing with your own shadow.

(b) Now look across the pond to the small archway structure (just visible in the picture above) beyond the pond near the left-hand side. Point a pencil at that arch from two different points on the Emmet sidewalk, measure the angles and the distance between the points, draw a scale diagram and deduce how far away the arch is from the Emmet sidewalk.

(c) Now go to the Lawn, the area on the same level as the Rotunda. The idea is to estimate the height of the Rotunda by drawing the right triangles. Notice from where I’m standing when I took this picture, some leaves are in line with the top of the Rotunda. I could find the

distance from where I’m standing to the base of the Rotunda (then I need to find what it is to the central point), then the distance from where I took the picture to the point directly under those leaves, and finally the height of the leaves. Or, I could look from ground level at a friend who walks to the point where the top of her head just blocks out the top of the Rotunda, and figure it from there.

2. How high does the Sun get in the sky at midday? (The high point is close to 1 pm since we’re on EST.) By “how high” I mean an angle; which I want you to measure, by finding an upright object (like holding a pencil vertical) and marking off its shadow (easier to do with two of you!), then drawing a triangle. Or, you could angle a pencil towards the Sun so it has no shadow, then measure the angle it is making with the vertical (or horizontal—just be clear which one you mean in writing this up).

Given that Charlottesville is 38 degrees from the Equator, what would that angle be on Midsummer’s day? On Midwinter’s day? (This is a lot easier to figure out if you have a globe to look at – try to find one!)

3. Some years ago, a very attractive photograph of the Rotunda viewed from the middle of the Lawn was on sale, it was night and the full Moon was visible quite close to the Rotunda, so the shape of the Rotunda echoed the Moon. How could you prove this photo was a fake?

4. You set up base camp on a bear hunting expedition. You travel exactly twenty miles south, rest for a meal, then go twenty miles east, where you shoot a bear. Now you go twenty miles north and you are back at your base camp. What color was the bear? Give your reasoning.

5. Plato defined a regular solid as one with every edge the same length, and the same number of edges meeting at each vertex. Explain in your own words (but with diagrams) why there can only be five such solids.

6. Prove that the square root of three cannot be expressed as a ratio of whole numbers. (Hint: remember that any whole number can be expressed as a product of prime factors. The proof of the square root of two was all about odd and even numbers, this time concentrate of factors of 3, not 2.)