PIT AND THE PENDULUM Did Edgar Allen Poe know His Math? Sharon Wiest

DO YOU KNOW THE STORY? (Do your students?) LET ME TELL YOU A STORY …….. You could start by going to you-tube: And choose either video (but not today) : Pit and the Pendulum Pit and Pendulum Rap Pit and the Pendulum (part 2 of 2) Vincent Price -- Pit & Pendulum part II

Details from the story: As the pendulum descends to within 3 inches, and is 10 to 12 swings away from starting to cut his clothes before cutting his body, the rats cut the ropes, and the main character is free to roll off the table.

MATH STANDARDS WRITE/CREATE A MATHEMATICAL EQUATION TO REPRESENT A GIVEN SITUATION DRAW A FIRST QUADRANT GRAPH IN THE COORDINATE PLANE TO REPRESENT INFORMATION IN A TABLE OR SITUATION REPRESENT A PROBLEM SITUATION, DESCRIBE THE PROCESS USED TO SOLVE THE PROBLEM, AND VERIFY TO REASONABLENESS OF THE SOLUTION

Extract and organize mathematical information from symbols, diagrams, and graphs to make inferences, draw conclusions and justify reasoning Make and test conjectures based on data (or information) collected from explorations and experiments Represent proportional relationships using graphs, tables, and equations Determine slope and y-intercept of a linear function for given table of data

Determine and justify whether a given verbal description represents a linear relationship Create a scatterplot for two-variable data set, and use trend lines to make predictions Extract information from graphps to make inferences and draw conclusions Represent a function four ways: symbolic expression, graph, table, verbally. Make connections among these representations Evaluate the argument and conclusion Verify accuracy in context of the original problem

TIME TO EXPLORE!

WHAT DATA DID WE COLLECT? (which order do you want to use?) Length of the pendulum OR Time for one complete swing (period) Time for one complete swing (PERIOD) OR Length

SLOPE Let’s calculate a couple of slopes using pairs of data by hand. Slope (m) = (y2 – y1) (x2 – x1) Using: 1. first two points 2. second and third points 3. first and third points 4. a set of points which pique your interest

OBSERVATIONS What units are we using? Is your first slope the same as others here? Are the slopes approximately the same for the first few points? If other people’s first slope is very different, are the two slopes reciprocals?

Comparing the slope calculations for several of the smallest data pairs – Do we think the relationship between period and length is linear? Let’s create a linear equation using one of our slope values: y– y1 = m(x – x1) What do we expect the time of a complete period to be if the length of the pendulum is 5 ft?

Let’s put our data in your calculator or in FATHOM Refresher for calculator: STAT/EDIT/ highlight L1/CLEAR/ENTER CLEAR LIST 2 also Enter data Quit STATPLOT/TURN PLOT 1 ON(or some other plot) Check to make sure that the lists named for x and y match your data ZOOM/ ZOOMSTAT

Does the last answer fit with our results? Let’s look at the scatterplot of the complete data set. Do we think really think this is a linear relationship?

Regression Analysis for Data that is not Linear How can we use the calculator or fathom to get a non-linear equation for the data? What equation do we get?

Let’s use our new equation Predict the time for a complete period for some pendulum length within the domain of our data (interpolation) How do this value fit with our data? HOW LONG DOES THE PRISONER (Mark) HAVE TO GET OFF THE TABLE ? Do we need to make some assumptions?

QUESTION: Has anyone else BEFORE us investigated the relationship between period and length of the pendulum? From Physics: Period of a Pendulum = 2*Pi * sqrt(length/gravity constant) T = 2π√(L/g) How does OUR equation compare to the accepted physics relationship?

Note: Units revisited trials and tribulations in trying to jog an old memory