MIT Blossoms lesson on Elasticity: studying how Solids change shape and size Handouts for students

Transcription

1 MIT Blossoms lesson on Elasticity: studying how Solids change shape and size Handouts for students Sourish Chakravarty Postdoctoral Associate The Picower Institute for Learning and Memory Massachusetts Institutes of Technology (MIT)

2 Spring Point mass Undeformed spring x

3 Deformation under tensile load F A x = δ x Reaction Force F R Applied Force F A

4 Deformation under compressive load F A x = δ x Reaction Force F R Applied Force F A

5 General case: F R δ n F R = kδ n k : Spring Constant (a measure of stiffness of the spring) Special case: n = 1 F R = kδ Hooke s law (linear spring)

6 From Newton s 2 nd Law of motion, F R F A = (mass)(acceleration) When acceleration is absent and/or mass is negligible, kδ n F A = 0 Or, F A = kδ n

7 Force vs. Deformation curve Applied Force, F A Slope of curve = k = h b measure of stiffness of the material Linear elasticity Constant slopes b h Example: Most materials under small deformation relative to undeformed configuration Change in length, δ

8 Force vs. Deformation curves for Linear and Nonlinear Elasticity Applied Force, F A Nonlinear case: Changing slopes. Example: Rubber under large relative deformation. Linear case: Constant slopes. Example: Most materials under small relative deformation Change in length, δ

9 Example of Activity 2 (Controlled extension of rubber band)

10 Beginning of experiment

11 Rubber band extended due to weight of coins During experiment

12 More coins added During experiment

13 Unloaded End of experiment

14 Loading sequence number, n Mass (kg) Reading from ruler, Y (metre) Extension, d= Y Y 0 (metre) Force = Mass*9.81 (newton) 0 0 Y

15 Loading sequence number, n Mass (kg) Reading from ruler, Y (metre) Extension, d= Y Y 0 (metre) Force = Mass*9.81 (newton)

16 Force, F (Newton) Example of Activity 2: Controlled extension of rubber band Force vs. Deformation plot for rubber band tensile test Extension, d (metre)

17 Example of Activity 3 (Launching match-stick using rubber-band as a projectile)

18 Force, F (Newton) Force vs. Deformation plot for rubber band tensile test z (in metre) Fitting nonlinear curve to data (Use computer or draw by eye) Extension, d (metre) Potential energy stored in the rubber band for extension of z metre (area under the Force vs. Deformation curve from d= 0 up to z metre)

19 Total elastic potential energy stored = area under Force vs. Deformation curve Observation Fitted curve Applied Force, F A Change in length, δ Approximating the area under the Force vs. Deformation curve as a sum of areas of multiple nonoverlapping trapezia/trapezoids Applied Force, F A Change in length, δ

20 Side-view l 2 l 2 Top-view Rubber-band before stretching H Side-view H Top-view After rubber-band is stretched by 2 inch Potential energy stored in rubber band Side-view H Rubber-band is released Projectile is launched (part of energy released is converted to kinetic energy of projectile) Projectile hits ground. R Point of first contact with ground

21 # Range, R (metre) Median

22 Kinetic energy of projectile at launch when rubber band is released Assuming: (1) Projectile launched horizontally, and (2) No air resistance, Kinetic energy of projectile = (Mass of projectile)(horizontal velocity of projectile at launch) 2 /2 (Horizontal velocity of projectile at launch) 2 = (Range of projectile)2 (Acceleration due to gravity) 2(Elevation oflaunch point) Kinetic Energy of Projectile at launch Potential Energy stored in Rubber Band < 1

23 Activity 3: Launching a match-stick using the rubber-band as a projectile Range, R (median of range values from 5 repetitions of projectile launch) Elevation of launch, H Acceleration due to gravity, g = 9.81 m/sec 2 Mass of projectile, M Velocity of launch of projectile, v = R g/2h Kinetic energy of projectile at launch, KE (Pr) = Mv 2 /2 Potential energy stored in rubber-band for z metre of extension, PE (Rb) in Joule Ratio = KE (Pr) /PE (Rb) =? DataAnalysisForActivities2and3.xlsx (sheet 2)