Christian Ucke

Technical University Munich

Content

1) Some physics toys as an introduction (without a long explanation)

2) Three toys with a more detailed explanation a) jumping animals and toys b) paper clip top c) 3D Chromadepth-Glasses

3) Literature and links for physics toys

Physics and Toys

10'

30'

5'

45'

N. Bohr and W. Paulilooking at a tippe-top (= flip-over top)

University of Lund31.05.1951

Calculated and animated behaviour of a tippe-top

Greek boy playing Yo-YoVase decoration 450 B.C.

Antikenmuseum Berlin

Physicist playing Yo-YoW. Bürger: Das Jojo, ein physikalisches SpielzeugPhys. Blätt. 39 (1993), 401-404

Climbing monkey The secret tube

principle of the siphon

Funfare (fanfare)

Paradoxical hourglass

liquid: saturated solution ofCaCl2 in water; 1.35g/cm3

balls: polystyrol; 1.05g/cm3

Paradoxical hourglass

2

mm

height of the jump h 0,5m

acceleration distance d 2mm

acceleration a = h·g/d = 2500ms-2

250g

(uniform acceleration assumed g = 10ms-2)

Pulex irritans (= human flea)

dhgaresults

advandghvFrom

/

22

jumping animals

height of the jump h = 1.2m (±10%)

=> Epot = mgh = 0.0145kg·10ms-2·1.2m = 0.17J

compressing on a scale

F 19N (= 1.9kg); d 3.5cm (better 3.2cm)

spring stiffness c = F/d 543Nm-1 (better 570Nm-1)

=> Espring = 0.5·c·d2 = 0.5·570Nm-1·0.0322m2 = 0.29J

initial acceleration a = F/m – g 190g

(m = head + rubber cup + 1/3spring) (a continuously decreasing!)

jumping toy

energyenergyenergy

kineticpotentialspring

vm

c

gmdgm

c

gmcd

c 21

111

2

12

222

11

11 8

ms

c

gmd

m

cv

Maximum velocity of the head

m 3

m 1

m 3 m 3m 3

m 1

m 1

d

m ·g/c1

0

jumping toy

frames of a digital video with 1000 pictures per second

Video2000 pictures/second146 pictures = 73 ms

jumping toy

jumping toyexperimental results

0,00 0,01 0,02 0,03 0,040

2

4

6

8

velocitymean

= 72Hz0.0139s

originalsmoothed

time [s]

ve

loc

ity

[m

/s]

0,00 0,01 0,02 0,03 0,040,00

0,05

0,10

0,15

0,20

Analysis jump3 (2000 pictures/s)

po

sit

ion

he

ad

[m

]

time [s]

jumping toy

calculation of the initial acceleration:

c = 570Nm-1; d = 0.032m; m = 0.0098kg; y = 0

gmsgm

ydcya 1851851

)()( 2

The acceleration continuously decreases

0,00 0,01 0,02 0,03 0,040

2

4

6

8

velocitymean

= 72Hz0.0139s

originalsmoothed

time [s]

ve

loc

ity

[m

/s]

0,00 0,01 0,02 0,03 0,04-3000

-2000

-1000

0

1000

2000

originalsmoothed

time [s]

ac

ce

lera

tio

n

[m/s

2]

Mechanical energy is lost in the rubber cup.

Several turns of the spring are compressed into the rubber cup;This part of the spring can decompressonly with strong friction.

jumping toy

jumping toy

Using calculus you may write down the differential equation for the starting process. This is the well-known equation of an oscillating mass hanging on a spring (harmonic oscillator), where damping is neglected 

 The solution with the initial conditions and y(0) = -d is

cygmym 11

0)0( y

t

m

c

c

gm d

c

gm ty

1

11 cos)(

and

t

m

c

c

gm d

m

c ty

1

1

1

sin)(

you get the maximum velocity)0062.0(2

1 sc

mt

For

T. Sakai: Topics on tops which enable anyone to enjoy himself, Mathematical Sciences (Surikagaki = ) 271, 18-26 (1986)

The paper-clip top (Sakai-top)

The paper-clip top (Sakai-top)

The paper-clip top (Sakai-top)

sin

2cos

11 2

1

s

rdrr

sxds

sx

center of gravity of the spokes

moment of the spokes cos2222 rxmM

= density per unit length

cos22

rx

sin2 2111 rxmM

center of gravity of the arc

moment of the arc

M1 = M2 tan = ½ or = 26.570 or ß = 2 = 53.130

Approximation

x1 = r·cos

M1 = m1·x1 = 2r2 ··sin·cos

x2 = r/2·cos

M2 = m2·x2 = r2 ··cos

M1 = M2 yields sin = ½ = 300 or = 600

The paper-clip top (Sakai-top)

unstable topfor h = 1.65·r

Ix < Iz < Iy

I = moment of inertia;

Iz = moment of inertia relative to the z-axis etc.

The paper-clip top (Sakai-top)

further possibilities

The paper-clip top (Sakai-top)

Landskonferansen om

fysikkundervisning

Chromatic Aberration in the human eye

chro

mat

ic a

bb

erra

tio

n

[d

pt]

tra

nsm

issi

on

[%

]

400 500 600 700

0

1

2

wavelength [nm]

chrom atic abberrationof the hum an eye

400 500 600 700

FF = fovea

0

5

0

100

wavelength [nm]P

opticalaxis

P = point of intersection optical axis - retina

lefteye

transm ission filter of cobaltg lass

Chromostereoscopyor

Colour Stereo Effect

im age depth apparent positionof the red source

red b luepoint light source

bluelightray

redlightray

redlightray

optica l ax isof the eye

bluelightray

F = Fovea cen tra lis

P = point o f in tersection optica l axis - re tina

F PFP

apparent positionof the b lue source

Amplification of theColour Stereo Effektby diaphragms

apparen t positiono f the red source

im age depth

red b luepo in t ligh t source

b lueligh tray

redligh tray

redligh tray

op tica l ax iso f the eye

b lueligh tray

D iaphragm

F PFP

apparen t positiono f the b lue source

Amplification of theColour Stereo Effektby prisms

im age depth

apparent positionof the blue source

red lightray

blue lightray

red lightray

blue lightray

optical axisof the eye

apparent positionof the red source

F P FP

prism s

red-blue point light source

Amplification of theColour Stereo Effektby blazed gratings

apparent positiono f the red sou rce

im age depth

C hrom aD ep th g ra tings

b lue ligh tray

red ligh tray

red ligh tray

optica l ax is o f the eye

b lue ligh tray

red-b lue po in t ligh t source

F PFP

apparent positiono f the b lue source

Blaze-Grating of the ChromaDepth-Glass

electron microscopepicture

B laze-conditions:d iffraction: = g ·sin and re fraction : n ·s in = s in (

Landskonferansen om

fysikkundervisning

URL: http://www.e20.physik.tu-muenchen.de/~cucke/

published by:American Associationof Physics Teachers

June 2000

~$35

Magic Egg