Pirate x and Lady y. Christopher Mundorff, Tamas Wiandt. November 21, 2013

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1 Introduction 1 of 15 Pirate x and Lady y Christopher Mundorff, Tamas Wiandt November 21, 2013

2 Introduction 2 of 15 Introduction Introduction Pirate x and Lady y The Exhibit Torque Definition, History, Connection to the Exhibit Experiments Numbers, Variables, Problems Puzzles Weight Measure Puzzles Extra with Torque An Interesting Result

3 The Exhibit Pirate X and Lady Y is the YouTube video of the exhibit. Can also be found here: Pirate x and Lady y 3 of 15

4 Introduction Pirate x and Lady y Torque Experiments Puzzles Extra with Torque Torque How can two kids of different weight play on the seesaw? Torque: τ= r F; magnitude: τ = r F sin θ; in our case: τ = r m g. Torque 4 of 15

5 History Archimedes: Give me a place to stand on, and I will move the Earth. Law of the Lever: Magnitudes are in equilibrium at distances reciprocally proportional to their weights. Torque 5 of 15

6 Balance = Equations The above idea means that balance is achieved when the weights multiplied by the lengths of the arms add up to the same number r r r r r r a b c d e f (3r)a + (2r)b + (r)c = (r)d + (2r)e + (3r)f 3a + 2b + c = d + 2e + 3f Torque 6 of 15

7 An Example = 2 7 Torque 7 of 15

8 Experiments 8 of 15 Just the Numbers Pick two different weights - can we always balance them? When can we balance them? Pick three different weights - can we always balance them?

9 Experiments 9 of 15 Variables Pirate x and Lady y - weights unknown. What is the easiest way to figure out their values? Set up an equation - then use balancing to simplify the equation.

10 Experiments 10 of 15 One- and Two-Step Equations Use only x or y here. Balance the ship with the unknown on the twoor three spot. First, replace 3x or 2x with three x s or two x s on the one spot. Do the same on the other side with the numbered weights. Then divide to get x. Two-step version: very similar, but put a weight on the left side as well; first, remove the numbered weights from both sides to maintain balance. Then back to case one.

11 Experiments 11 of 15 Tougher Problems Use both x and y - how many measurements do we need to identify them? Set up a system of equations, then solve it.

12 Puzzles 12 of 15 Weight Measure Puzzles We have 10 bags with 1000 coins each. In one of the bags, all coins are forgeries. A true coin weighs 1 gram; each counterfeit coin weighs 1.1 gram. If we have an accurate scale, which we can use only once, how can we identify the bag with the forgeries? On a Christmas tree there were two blue, two red, and two white balls. All seemed same. However, in each color pair, one ball was heavier. All three lighter balls weighed the same, just like all three heavier balls. Using a two-pan balance scale twice, identify the lighter balls. There are 9 similar balls. Eight of them weigh the same and the ninth is a bit heavier. How would you identify the heavier ball if you could use a two-pan balance scale only twice?

13 Puzzles 13 of 15 Puzzles with Nested Balances Use each of the numbers 1, 2,..., n exactly once. n = 5: Check out efriedma/weight/ for more.

14 Hydrostatic Pressure and Force Consider a rectangular tank, filled with ρ density fluid. The hydrostatic pressure at depth h is given by P = ρgh. If the tank is filled to height h, the force exerted on a side of a tank with area A is F = ρgh A/2. Extra with Torque 14 of 15

15 A Strange Aquarium a c m b When the water is at rest, the tank does not rotate: (a/2)ρgh(am)/2 + (b/2)ρgh(bm)/2 = (c/2)ρgh(cm)/2 a 2 + b 2 = c 2 Extra with Torque 15 of 15

Torque and Static Equilibrium

Torque and Static Equilibrium Rigid Bodies Rigid body: An extended object in which the distance between any two points in the object is constant in time. Examples: sphere, disk Effect of external forces