Investigate the hoop`s motion.” 1

advertisement
1
5. LOADED HOOP
“Fasten a small weight to the inside of a hoop and set
the hoop in motion by giving it an initial push.
Investigate the hoop’s motion.”
2
Example of motion - video
3
0,10
0,08
0,06
0,10
0,02
0,08
0,00
0,06
-0,02
0,04
-0,06
-0,08
0,0
0,02
Center of mass
Center of hoop
14
0,000,5
12
1,5
1,0
X [m]
-0,02
-0,04
-0,06
2,0
10
Angle [rad]
-0,04
Y [m]
Y [m]
0,04
8
6
Center of mass
Center of hoop
4
-0,08
2
0,0
0,5
1,0
0
0,0
1,5
X [m]0,2
0,4
2,0
0,6
Time [s]
0,8
1,0
4
Outline
Experimental approach
• Hoop properties
• Giving initial push
• Parameters
Theoretical modeling
• Kinematics and dynamics
• Modes of motion
Comparison of experimental data and theory
• Modes of motion
• Dependence of angular frequency on angle
for different weight mass
• Hop analysis
• Energy analysis
5
Hoop properties
29,5 cm
99 mm
ℎ = 24.4
ℎ = 186 
6
Hoop properties
Hoop 1
Hoop 2
Material
Stryofoam
Wood
Shape
Toroidal
Full ring
Determing
moment of inertia
Mathematicaly
Eksperimentaly
 =
−
Δ
2
2
+
3 Δ
4 2
Determing
Mathematicaly

distance from
 =


center of the mass
2
 +   2
0
 = 2

 = 0 − 
Eksperimentaly
-hoop was stabilised
on screwdriver
7
Experimental aproach
• Giving initial push
 Incline
Frame for centering and
releasing the hoop
 Colision
• Method of motion recording
 3 small areas marked with reflective stickers
 Motion recorded in 120 fps
 Stickers tracked in program for video analysis ImageJ
8
Theoretical modeling

weight

ℎ − hoops mass
 − mass of the weight
 – moment of inertia around center
 − friction force
 − normal force
 − center of the mass distance
,  − coordinates of the hoops center
 Mg
N
F
Kinematics of the hoop:
 =  +  
 =  + ()
Dynamics of the hoop:
 = 
 =  − 
  =    − 
 = ℎ +  

 = 2
  
=
=


∆R
r
R
9
Modes of motion
• Rolling
• Rolling with slipping
• Hop
10
Rolling
• Properties of rolling
  = ,  = ,  = 
  > 0,  = 0,  = 0,  = 0
  < 
• Solving equations for dynamics with assumption of rolling
we obtain:
 =
+0
+
2

1+
=


0 2 +  − 
sin2 

+



= 1−



=  −  2 + 
− 2
1+
+

+
11
Rolling with slipping
• Properties of rolling with slipping
  = −

| |
whereby  is relative velocity between hoop on the contact
of the surface and the surface
  = 0,  = 0,  = 0,  = 0
• Two cases:
  > 
 The hoop is rotating more than it’s translating
  < 
 The hoop is translating more than it’s rotating
12
Hop
• The center of mass moving along a parabola with
constant speed in the x direction
• Conservation of angular momentum
• Most probable hop when weight is on highest position
•=0
13
Determing hoop motion - 1
Rolling condition
•  = 
 

Hop
()
Hop
•  − 
14
Determing hoop motion - 1
Rolling condition:
 < 


Rolling with sliping
condition:
 > 

||


| |
Hop:
=0
15
Hop analysis - 2
30
-1
[rad s ]
25
0,06
20
15 0,05
0,04
0,04
Y [m]
10
0,03
Y[m]
0,0
0,02
0,5
1,0
0,02
1,5
2,0
X [m]
0,01
0,00
0,00
Center of hoop
-0,01
0,0
0,5
1,0
1,5
X [m]
0,8 2,0
0,9
1,0
1,1
X [m]
1,2
1,3
1,4
16
Analysis of motion without hop - 3
Rolling condition
•  = 
17
Analysis of motion without hop - 3
• The motion of the hoop
can be considerated as
rolling  < 






= 1
sin2 
−
+
1+
+

+
− 2
=  −  2 + 
18
Conservation of energy
Energy of a hoop:
 1 2
=  +  
 2 
2
+ 
19
Dependence of angular velocity on angle
for different weight mass
  =
32
4598.73
3.2 + 
2
− 79.11
weight 26,19g -theory
weight 26,19g - exp
weight 43,75g - theory
weight 43,75g - exp
no weight - exp
no weight - theory
30
28
26
-1
[rad s ]
24
22
  =
20
4620,46
− 79,11
3,49 + cos 
18
16
m [g]

14
43,75
3.2
26,19
3.49
0
→∞
  = 15,65
12
10
0
2
4
6
[rad]
8
=
10
+0 2
+
12


0 2 +  −  ,  =
1+

20
Conclusion
• Three modes of motion were observed
 Rolling
 Rolling with sliping
 Hop
• We analyticaly solved equation of motion for rolling case
• =
+0 2
+
0 2 +




− ,=
1+

and the solution fits the experimental data
• When the hoop hop its center of mass will move on parabolic trajectory and it
will have constant angular momentum
21
Reference
• M. F. Maritz, W.F.D. Theron; Experimental verification of
the motion of a loaded hoop; 2012 American journal of
physics
• M. F. Maritz, W.F.D. Theron; The amazing variety of
motions of a loaded hoop; Mathematical and Computer
Modelling 47 (2008) 1077-1088
• Lisandro A. Raviola, O. Zarate, E. E. Rodriguez; Modeling
and experimentation with asymmetric rigid bodies: a
variation on disks and inclines; March 31, 2014
• W. F. D. Theron, Analysis of the Rolling Motion of Loaded
Hoops, dissertation, March 2008
22
IYPT 2014
CROATIAN TEAM
THANK YOU
Reporter: Domagoj Pluščec
23
Hoop1
n
 [g]
M[g]
I[kgm^2]∙ 10−4

k

1
17,8
42,2
4,33
0,337
0,657
4,92
2
26,2
50,6
5,17
0,414
0,654
4,01
3
35,5
59,9
6,1
0,474
0,652
3,49
4
44
68,4
6,95
0,515
0,65
3,2
24
Determing hoops motion – another
example
25
Rotational velocity in time – another
example
26
Energy – another example
Download
Related flashcards

Statistical mechanics

17 cards

Special relativity

26 cards

Classical mechanics

25 cards

Mechanics

22 cards

Statistical mechanics

17 cards

Create Flashcards