July 16, 2007COSMOS @ UCSD - R.A. de Callafon Dynamics of Moving Objects in Kinetic Sculpture (Ball Drop Physics I) Raymond de Callafon Dynamic Systems

July 16, 2007COSMOS @ UCSD - R.A. de Callafon

Dynamics of Moving Objects in Kinetic Sculpture

(Ball Drop Physics I)

Raymond de CallafonDynamic Systems & Control Group

Center of Magnetic Recording ResearchUCSD, Dept. of MAE

Email: [email protected]

COSMOS LECTURE

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


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 …


Vertical Drop of BallDynamics can be described by Newton’s 2nd law:

Force F(t) is gravitation: F = – Mg

is constant and gives Newton’s law:

Solution for y(t):

Simplest Model of Ball Physics

)()()(2

2

tydt

dMtMatF

gtydt

dty

dt

dMMg )()(

2

2

2

2

2

2

1)( gtty

M

y(t)

F(t)



Vertical Drop of BallSolution:

indicates that position y(t) changeswith a parabolic function, whereas:

velocity:

acceleration:

2

2

1)( gtty

gttydt

dty )()(

gtydt

dty )()(

2

2

M

y(t)

F(t)



Vertical Drop of Ball

Position:

Velocity:

Acceleration:

2

2

1)( gtty

gttydt

dtv )()(

gtydt

dta )()(

2

2


Vertical Drop of BallWith 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 ithits the ground at height h ?


2

2

1)( gtty

M y(t)

F(t) h

ghtgth 2

2

1 2

hgghggttv 22)(


Vertical Drop of BallIf you are only interested in thevelocity of the ball due to height difference(typical for Kinetic Sculpture)then solution easy tofind with conservation of energy:

potential + kinetic energy = constantorpotential energy = kinetic energy


M y(t)

F(t) h

hgv 2


End Velocity of Ball Droppotential energy = kinetic energy potential energy P of mass M height h:

kinetic energy E of mass M with velocity v:

conservation of energy yields:


M y(t)

F(t) h

hgvMvMghEP 22

1 2

MghP

2

2

1MvE


Ball Drop ExperimentBased on velocity datashown on the right: With

what is the height h fromwhich the ball is dropped?


21081.9 m/s g

m g

vh hgv

2

5102

10

22

2

m gtty 51102

1

2

1)( 22

Alternative solution:


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:


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?

hgvMvMghEP 22

1 2

h


Ball on a Ramp

Difference between Skidding and Rolling

Potential energy P converted in 2 types of kinetic energy Ev and Er : P = Ev+ Er where

1. Kinetic energy Ev for (horizontal) velocity of ball

2. Kinetic energy Er for rolling ball


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)

2

2

1vMEV

2

2

1wIER


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!


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:

2

5

2MrI

r

vw


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:

22

222

5

1

5

2

2

1

2

1vM

r

vMrwIER

222

10

7

5

1

2

1MvMvMvEEE RV


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

hgv MvMgh EP7

10

10

7 2

MghP

222

10

7

5

1

2

1MvMvMvEEE RV

Note: 10/7 is smaller than 2!


Boll drop lab

Experimental verification of vertical ball drop

Build your own experimentalapparatus

We use one of the optical sensorsand microprocessor controlbox to measure velocity

Compare theory

with experiments

hgvMvMghEP 22

1 2


Boll drop lab

Experimental verification of inclined ball velocity

Build your own experimentalapparatus

We use one of the optical sensorsand microprocessor controlbox to measure velocity

Compare theory

with experiments

hgv MvMgh EP7

10

10

7 2

Documents

July 16, 2007COSMOS @ UCSD - R.A. de Callafon Dynamics of Moving Objects in Kinetic Sculpture (Ball Drop Physics I) Raymond de Callafon Dynamic Systems