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Introduction to Linearization ( No units, no uncertainties, just the core idea )

Mr. Klapholz Shaker Heights High School. Introduction to Linearization ( No units, no uncertainties, just the core idea ). The purpose of linearization is to get the equation that describes real data. A scientist varies the mass, and measures the acceleration. Force is kept constant.

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Introduction to Linearization ( No units, no uncertainties, just the core idea )

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  1. Mr. Klapholz Shaker Heights High School Introduction to Linearization(No units, no uncertainties, just the core idea) The purpose of linearization is to get the equation that describes real data.

  2. A scientist varies the mass, and measures the acceleration. Force is kept constant.

  3. What shape will we seewhen we graph it? Acceleration Mass

  4. The greater the Mass, the less the acceleration Acceleration Mass

  5. Acceleration It is tough to know the equation of this function. a = ? Mass

  6. Acceleration So we linearize it. Mass

  7. We guess that Acceleration = k / Mass

  8. We guess that acceleration = k / Mass

  9. We guess that acceleration = k / Mass

  10. What shape will we seewhen we graph it? Acceleration 1 / Mass

  11. Acceleration 1 / Mass

  12. Acceleration 1 / Mass

  13. y = mx +b Acceleration 1 / Mass

  14. a = (slope)×(1/Mass) + b Acceleration y = mx +b 1 / Mass

  15. Find the slope and the intercept.

  16. Slope Slope = Rise / Run Slope = Da / D(1/M) Slope = ( 12 – 3 ) / (1 – 0.75) Slope = 9 / 0.75 Slope = 12

  17. Intercept Since the graph goes through the origin, the intercept is 0.

  18. So, what is the equation?

  19. a = ?

  20. a = 12 (1/M) • Notice that we were able to write down the conclusion to the lab only because we had linearized the data. • The function could be said to be “linear in 1/M”. • But what we really wanted was the function, and we have it: a = 12 / M. • FYI: Newton’s second law says, in part, that acceleration = Force / mass.

  21. Our last example…

  22. A researcher changes the distance that a spring is compressed, and measures the energy in the spring.

  23. What shape will we seewhen we graph it? Energy = ? Energy Distance

  24. Energy Distance

  25. The greater the Distance, the greater the acceleration. Energy Distance

  26. Energy It is tough to know the equation of this function. E = ? Distance

  27. Energy Let’s linearize it. Distance

  28. We guess that E = k × D2

  29. We guess that E = k × D2

  30. We guess that E = k × D2

  31. What shape will we seewhen we graph it?

  32. Energy Distance

  33. Energy Distance

  34. y = mx +b Energy Distance

  35. a = (slope)×(1/Mass) + b Energy y = mx +b Distance

  36. Find the slope and the intercept.

  37. Slope Slope = Rise / Run Slope = DE / D(D2) Slope = ( 32 – 2 ) / ( 16 – 1 ) Slope = 30 / 15 Slope = 2

  38. Intercept Since the graph goes through the origin, the intercept is 0.

  39. So, what is the equation?

  40. E = ?

  41. E = 2 D2 • The data indicate that the energy stored in a spring is proportional to the square of the compression distance.

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