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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
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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
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Hoop properties
29,5 cm
99 mm
𝑚ℎ = 24.4𝑔
𝑚ℎ = 186 𝑔
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Hoop properties
Hoop 1
Hoop 2
Material
Stryofoam
Wood
Shape
Toroidal
Full ring
Determing
moment of inertia
Mathematicaly
Eksperimentaly
𝐼𝑐𝑚 =
𝑅−
Δ𝑅
2
2
+
3 Δ𝑅
4 2
Determing
Mathematicaly
𝑚𝑢
distance from
𝑟𝑐𝑚 =
𝑟
𝑀
center of the mass
2
𝑚𝑜 + 𝑚𝑢 𝑟 2
𝐼0
𝑇 = 2𝜋
𝑚𝑔𝑑
𝐼𝑐𝑚 = 𝐼0 − 𝑚𝑔𝑑
Eksperimentaly
-hoop was stabilised
on screwdriver
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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
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Theoretical modeling
𝑦
weight
𝜃
𝑚ℎ − hoops mass
𝑚𝑤 − mass of the weight
𝐼 – moment of inertia around center
𝐹 − friction force
𝑁 − normal force
𝛾𝑅 − center of the mass distance
𝑥, 𝑦 − coordinates of the hoops center
𝛾𝑅 Mg
N
F
Kinematics of the hoop:
𝑥𝑐 = 𝑥 + 𝛾𝑅𝑠𝑖𝑛 𝜃
𝑦𝑐 = 𝑦 + 𝛾𝑅𝑐𝑜𝑠(𝜃)
Dynamics of the hoop:
𝑀𝑥𝑐 = 𝐹
𝑀𝑦𝑐 = 𝑁 − 𝑀𝑔
𝐼 𝜃 = 𝑚𝑤 𝑔𝑟𝑠𝑖𝑛 𝜃 − 𝐹𝑅
𝑀 = 𝑚ℎ + 𝑚 𝑢
𝑥
𝐼 = 𝑘𝑀𝑅2
𝑚𝑤 𝑟 𝑟𝑐𝑚
𝛾=
=
𝑀𝑅
𝑅
∆R
r
R
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Modes of motion
• Rolling
• Rolling with slipping
• Hop
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Rolling
• Properties of rolling
 𝑥 = 𝑅𝜃, 𝑥 = 𝑅𝜔, 𝑥 = 𝑅𝛼
 𝑁 > 0, 𝑦 = 0, 𝑦 = 0, 𝑦 = 0
 𝐹 < 𝜇𝑁
• Solving equations for dynamics with assumption of rolling
we obtain:
 𝜔=
𝜂+𝑐𝑜𝑠𝜃0
𝜂+𝑐𝑜𝑠𝜃
2
𝑔
1+𝑘
𝜂=
𝛾
𝑔
𝜔0 2 + 𝑅 − 𝑅
sin2 𝜃
𝛾
𝜂+𝑐𝑜𝑠𝜃

𝑁
𝑀
=𝑔 1−

𝐹
𝑀
= 𝑔𝛾𝑠𝑖𝑛𝜃 − 𝑘 𝑅𝜔2 + 𝑔
− 𝛾𝑅𝜔2
1+𝜂𝑐𝑜𝑠𝜃
𝜂+𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
𝜂+𝑐𝑜𝑠𝜃
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Rolling with slipping
• Properties of rolling with slipping
 𝐹 = −𝜇𝑁
𝑣𝑟
|𝑣𝑟 |
whereby 𝑣𝑟 is relative velocity between hoop on the contact
of the surface and the surface
 𝑁 = 0, 𝑦 = 0, 𝑦 = 0, 𝑦 = 0
• Two cases:
 𝑅𝜔 > 𝑥
 The hoop is rotating more than it’s translating
 𝑅𝜔 < 𝑥
 The hoop is translating more than it’s rotating
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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
•𝑁=0
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Determing hoop motion - 1
Rolling condition
• 𝑥 = 𝑅𝜔
𝑣 𝑡
𝑅
Hop
𝜔(𝑡)
Hop
• 𝜔 − 𝑐𝑜𝑛𝑠𝑡
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Determing hoop motion - 1
Rolling condition:
𝐹 < 𝜇𝑁
𝜇𝑁
𝑀
Rolling with sliping
condition:
𝐹 > 𝜇𝑁
𝐹
|𝑀|
𝐹
𝑀
| |
Hop:
𝑁=0
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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
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Analysis of motion without hop - 3
Rolling condition
• 𝑥 = 𝑅𝜃
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Analysis of motion without hop - 3
• The motion of the hoop
can be considerated as
rolling 𝐹 < 𝜇𝑁


𝑁
𝑀
𝐹
𝑀
=𝑔 1
sin2 𝜃
−𝛾
𝜂+𝑐𝑜𝑠𝜃
1+𝜂𝑐𝑜𝑠𝜃
𝜂+𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
𝜂+𝑐𝑜𝑠𝜃
− 𝛾𝑅𝜔2
= 𝑔𝛾𝑠𝑖𝑛𝜃 − 𝑘 𝑅𝜔2 + 𝑔
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Conservation of energy
Energy of a hoop:
𝐸𝑡 1 2
= 𝑣 + 𝑘𝑐 𝜔𝑅
𝑀 2 𝑐
2
+ 𝑔𝑦𝑐
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Dependence of angular velocity on angle
for different weight mass
𝜔 𝜃 =
32
4598.73
3.2 + 𝑐𝑜𝑠𝜃
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
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28
26
-1
[rad s ]
24
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𝜔 𝜃 =
20
4620,46
− 79,11
3,49 + cos 𝜃
18
16
m [g]
𝜼
14
43,75
3.2
26,19
3.49
0
→∞
𝜔 𝜃 = 15,65
12
10
0
2
4
6
[rad]
8
𝜔=
10
𝜂+𝑐𝑜𝑠𝜃0 2
𝜂+𝑐𝑜𝑠𝜃
12
𝑔
𝑔
𝜔0 2 + 𝑅 − 𝑅 , 𝜂 =
1+𝑘
𝛾
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Conclusion
• Three modes of motion were observed
 Rolling
 Rolling with sliping
 Hop
• We analyticaly solved equation of motion for rolling case
• 𝜔=
𝜂+𝑐𝑜𝑠𝜃0 2
𝜂+𝑐𝑜𝑠𝜃
𝜔0 2 +
𝑔
𝑅
𝑔
𝑅
− ,𝜂=
1+𝑘
𝛾
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
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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
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IYPT 2014
CROATIAN TEAM
THANK YOU
Reporter: Domagoj Pluščec
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Hoop1
n
𝑚𝑢 [g]
M[g]
I[kgm^2]∙ 10−4
𝛾
k
𝜂
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
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Determing hoops motion – another
example
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Rotational velocity in time – another
example
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Energy – another example