1. July 16, 2007COSMOS @ UCSD - R.A. de Callafon Dynamics of Moving Objects in Kinetic Sculpture (Ball Drop Physics I) Raymond de Callafon Dynamic Systems & Control Group Center of Magnetic Recording Research UCSD, Dept. of MAE Email: callafon@ucsd.edu COSMOS LECTURE

2. 2 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Physics in Kinetic Sculpture Observations: main force acting on balls: gravity pendulums create “randomness” conversion of potential energy to kinetic energy

3. 3 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Energy in Kinetic Sculpture Energy Conservation: Motor + “ball elevator” increases potential energy Ball rolling down slides converts potential energy into kinetic energy Energy loss during conversion due to: Friction of balls on slides Loss of energy while balls bounce (trampoline or in baskets) Friction of air while balls move …

4. 4 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Vertical Drop of Ball Dynamics can be described by Newton’s 2 nd law: Force F(t) is gravitation: F = – Mg is constant and gives Newton’s law: Solution for y(t): Simplest Model of Ball Physics M y(t) F(t)

5. 5 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Simplest Model of Ball Physics Vertical Drop of Ball Solution: indicates that position y(t) changes with a parabolic function, whereas: velocity: acceleration: M y(t) F(t)

6. 6 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Simplest Model of Ball Physics Vertical Drop of Ball Position: Velocity: Acceleration:

7. 7 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Vertical Drop of Ball With solution: consider the following questions: 1. How long does it take (at what time t) to reach the ground at height h ? 2. What is the speed of the ball when it hits the ground at height h ? Simplest Model of Ball Physics M y(t) F(t) h

8. 8 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Vertical Drop of Ball If you are only interested in the velocity of the ball due to height difference (typical for Kinetic Sculpture) then solution easy to find with conservation of energy: potential + kinetic energy = constant or potential energy = kinetic energy Simplest Model of Ball Physics M y(t) F(t) h

9. 9 July 16, 2007COSMOS @ UCSD - R.A. de Callafon End Velocity of Ball Drop potential energy = kinetic energy potential energy P of mass M height h : kinetic energy E of mass M with velocity v : conservation of energy yields: Simplest Model of Ball Physics M y(t) F(t) h

10. 10 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball Drop Experiment Based on velocity data shown on the right: With what is the height h from which the ball is dropped? Simplest Model of Ball Physics Alternative solution:

11. 11 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Bouncing and Skidding Ball In Kinetic Sculpture most balls operate on a ramp Analysis of conservation of energy still holds! What matters is height difference! Illustration:

12. 12 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp End Velocity for Skidding Ball Velocity on the basis of energy conservation: where h = height difference: Important assumption: kinetic energy is only due to (horizontal) velocity of ball. What if ball is rolling? h

13. 13 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Difference between Skidding and Rolling Potential energy P converted in 2 types of kinetic energy E v and E r : P = E v + E r where 1. Kinetic energy E v for (horizontal) velocity of ball 2. Kinetic energy E r for rolling ball

14. 14 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Difference between Skidding and Rolling Kinetic energy due to ball velocity (as before): where M is mass of ball and v is velocity of ball Kinetic energy due to ball rotation: where I is inertia of ball and w is rotational velocity of the ball (in radians / second)

15. 15 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Difference between Skidding and Rolling We can actually compute the I = inertia of ball and w = rotational velocity of the ball!

16. 16 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Difference between Skidding and Rolling We can actually compute the I = inertia of ball and w = rotational velocity of the ball! Facts: Inertia I for a solid sphere (ball): Rotational velocity of ball with radius r:

17. 17 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Ball on a Ramp Difference between Skidding and Rolling Combining these facts we can compute kinetic energy due to the ball rotation: Total energy of rolling ball:

18. 18 July 16, 2007COSMOS @ UCSD - R.A. de Callafon h End Velocity of a Rolling Ball potential energy P of mass M at height h : kinetic energy E of rolling M with velocity v : conservation of energy yields end velocity of rolling ball: Ball on a Ramp Note: 10/7 is smaller than 2!

19. 19 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Boll drop lab Experimental verification of vertical ball drop Build your own experimental apparatus We use one of the optical sensors and microprocessor control box to measure velocity Compare theory with experiments

20. 20 July 16, 2007COSMOS @ UCSD - R.A. de Callafon Boll drop lab Experimental verification of inclined ball velocity Build your own experimental apparatus We use one of the optical sensors and microprocessor control box to measure velocity Compare theory with experiments