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
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 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
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)
5 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
Simplest Model of Ball Physics
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)
6 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
Simplest Model of Ball Physics
Vertical Drop of Ball
Position:
Velocity:
Acceleration:
2
2
1)( gtty
gttydt
dtv )()(
gtydt
dta )()(
2
2
7 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
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 ?
Simplest Model of Ball Physics
2
2
1)( gtty
M y(t)
F(t) h
ghtgth 2
2
1 2
hgghggttv 22)(
8 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
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
Simplest Model of Ball Physics
M y(t)
F(t) h
hgv 2
9 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
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:
Simplest Model of Ball Physics
M y(t)
F(t) h
hgvMvMghEP 22
1 2
MghP
2
2
1MvE
10 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
Ball Drop ExperimentBased on velocity datashown on the right: With
what is the height h fromwhich the ball is dropped?
Simplest Model of Ball Physics
21081.9 m/s g
m g
vh hgv
2
5102
10
22
2
m gtty 51102
1
2
1)( 22
Alternative solution:
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 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?
hgvMvMghEP 22
1 2
h
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 Ev and Er : P = Ev+ Er where
1. Kinetic energy Ev for (horizontal) velocity of ball
2. Kinetic energy Er for rolling ball
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)
2
2
1vMEV
2
2
1wIER
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 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:
2
5
2MrI
r
vw
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:
22
222
5
1
5
2
2
1
2
1vM
r
vMrwIER
222
10
7
5
1
2
1MvMvMvEEE RV
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
hgv MvMgh EP7
10
10
7 2
MghP
222
10
7
5
1
2
1MvMvMvEEE RV
Note: 10/7 is smaller than 2!
19 July 16, 2007COSMOS @ UCSD - R.A. de Callafon
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