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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