physics 1710—warm-up quiz two 2.0 kg disks, both of radius 0.10 m are sliding (without friction)...
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Physics 1710Physics 1710—Warm-up Quiz—Warm-up QuizTwo 2.0 kg disks, both of radius 0.10 m are sliding (without friction) and rolling, respectively, down an incline. Which will reach the bottom first?
1 2 3
37%
23%
40%
1. Rolling disk wins.
2. Sliding disk wins.
3. Tie.
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21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Solution:Solution:
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Kinetic Energy of sliding disk:
K1 = ½ mv2 = mg(h-z); v =√[2g(h-z)]
Kinetic Energy of rolling disk:
K2 = ½ mv2+ ½ I ω2= mg(h-z)= ½ mv2+ ½ ( ½ mr2) (v/r)2
= 3/4 mv2; v =√[4/3 g(h-z)]
Slider wins!
Consider two spindles rolling down a Consider two spindles rolling down a ramp:ramp:
a
b
c
c
Which one will win and why?
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
No Talking!No Talking!Think!Think!
Confer!Confer!
Peer Instruction Peer Instruction TimeTime
Which one will win and why?Which one will win and why?
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
11′′ Lecture Lecture •The energy of rotation The energy of rotation
K = ½ I K = ½ I ⍵⍵ 2 2
• Torque (“twist”) is the vector product of the Torque (“twist”) is the vector product of the “moment” “moment” and a force. and a force. τ = τ = r x Fr x F
• τ τ = I = I ⍺ = I d⍵/dt⍺ = I d⍵/dt
•Angular momentum Angular momentum LL is the vector product of the is the vector product of the moment arm and the linear momentum. moment arm and the linear momentum. LL= = r x p.r x p.
• τ = d τ = d LL/dt/dt
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Moment of Inertia—sphereMoment of Inertia—sphere
I = ∭ R I = ∭ R 2 2 ρ d ρ d VVN.B. : R N.B. : R 2 2 = r = r 22 – z – z 22
R R 2 2 = r = r 22 – (r cos – (r cos θ)θ) 22
=r =r 22 (1– cos (1– cos 2 2 θ)θ)
I = I = ∫∫0022ππ d dφφ∫∫00
ππ (1– cos (1– cos 2 2 θ)θ) sin sin θdθθdθ∫∫00aa r r 4 4 ρρdrdr
= [2= [2π][2- 1/3(2)][1/5 aπ][2- 1/3(2)][1/5 a55] ρ = ] ρ = [4[4π][2/3][1/5 aπ][2/3][1/5 a55] ] ρ ρ
==[4[4π/3][2/5 aπ/3][2/5 a55] ρ = 2/5 M a] ρ = 2/5 M a22
Rrr
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Kinetic Energy of Rotation:Kinetic Energy of Rotation:
K = ½ K = ½ ΣΣi i mmi i vvi i 22
K = ½ K = ½ ΣΣi i mmi i (R(Rii ω ωii ) ) 22
K = ½ K = ½ ΣΣi i mmi i R R ii 22ωωii 22
For rigid body ωFor rigid body ωii = ω = ω
K = ½ [K = ½ [ΣΣi i mmi i R R ii 22] ω ] ω 22
K = ½ K = ½ I ω I ω 22
With I =ΣWith I =Σi i mmi i R R ii 22 = the moment of inertia. = the moment of inertia.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Why do round bodies roll down Why do round bodies roll down slopes?slopes?
The torque is the “twist.”
θ Fsinθ
θFg
F = m a = mr F = m a = mr dω/dtdω/dt
rF = rrF = rFFg g sinsinθθ = mr= mr22dω/dtdω/dt
Torque = r x F = I αTorque = r x F = I α
τ τ = = r x Fr x F,, ||τ τ || = = rFrFsinsinθθ
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Torque and the Right Hand Rule:Torque and the Right Hand Rule:
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
rr
FF r r x x FF
X
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Vector Product:Vector Product:
C = A x BC = A x B
CCxx = = AAyy B Bzz – A – Azz B Byy
Cyclically permute: (xyz), (yzx), (zxy)Cyclically permute: (xyz), (yzx), (zxy)
||CC| =| =√[√[CCxx22 + C + Cyy
22 + C + Czz22 ]]
= = AB sin AB sin θθ
Directed by RH Rule.Directed by RH Rule.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Vector Product:Vector Product:A x B = - B x AA x B = - B x A
A x ( B + C ) = A x B + A x CA x ( B + C ) = A x B + A x Cd/dt d/dt ( ( A x BA x B ) = d ) = d AA /dt /dt x B + A x x B + A x dd B B/dt/dt
i x i = j x j = k x k = 0i x i = j x j = k x k = 0i x j = - j x i = ki x j = - j x i = k
j x k = - k x j = ij x k = - k x j = ik x i = - i x k = jk x i = - i x k = j
Torque Bar:Torque Bar:
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
rr
FF
τ τ == r r x x FFττ
A B C
Teeter-totter:Teeter-totter:
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
τ τ == r r x x FF
FF22
FF11
Where should the fulcrum be place to balance the teeter-totter?
Where should the fulcrum be place to balance the teeter-totter?
A B C
6%
86%
8%
A.A.
B.B.
C.C.
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101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Torque LadderTorque Ladder
Which way will the torque ladder move?
?
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Which way will the torque ladder move?
A B C
3%
27%
70%
A.A. Clockwise Clockwise B.B. CounterclockwiseCounterclockwiseC.C. Will stay balancedWill stay balanced
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101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Torque LadderTorque Ladder
Which way will the torque ladder move?
?
r
r sin θ
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Second Law of MotionSecond Law of Motion
L = r L = r xx pp is the “angular momentum.”
FF = m = m aa
Or Or F F = d= dpp/dt/dt
Then:Then:
r x r x F F = d (= d (r r xx pp)/dt)/dt
Torque = Torque = ττ = d = d LL//dtdt
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Angular Momentum:Angular Momentum:
L = r x pL = r x p
The angular momentum is the The angular momentum is the vector vector productproduct of the moment arm and the linear of the moment arm and the linear momentum.momentum.
∑ ∑ T = T = d d LL/dt/dt
The net torque is equal to the time rate of The net torque is equal to the time rate of change in the angular momentum.change in the angular momentum.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Angular Momentum:Angular Momentum:
Proof:Proof:∑ ∑ T = T = r x r x ∑∑FF = = r x r x d d pp/dt/dt
AndAndd d LL/dt = d( /dt = d( r x pr x p) /dt ) /dt
= d = d rr/dt/dt x p + r x x p + r x d d pp/dt./dt.
But But p = p = m d m d rr/dt/dt , , therefore therefore d d rr/dt/dt x p = 0 x p = 0
d d LL/dt = /dt = r x r x d d pp/dt/dtAnd thusAnd thus
∑ ∑ T = T = d d LL/dt./dt.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Second Law of MotionSecond Law of Motion
L = L = constant meansconstant means angular momentum si conserved.
Torque = Torque = ττ = d = d LL//dtdt
If If ττ = 0, then = 0, then
LL is a constant. is a constant.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Rotating Platform Rotating Platform DDemonstrationemonstration
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Analysis:Analysis:
•Why does an ice skater increase her angular Why does an ice skater increase her angular velocity without the benefit of a torque?velocity without the benefit of a torque?
L = r x pL = r x p= r x = r x ( m ( m vv))
= = r x r x ( m ( m r x r x ⍵⍵))
LLii = m = mii r rii 2 2 ⍵ ⍵
LLzz = (∑ = (∑ii m mii r rii 2 2 ) ⍵ ) ⍵
LLzz = I ⍵; & ⍵ = L = I ⍵; & ⍵ = Lzz / I / I
Therefore, a decrease in I ( by reducing r) will Therefore, a decrease in I ( by reducing r) will result in an increase in ⍵.result in an increase in ⍵.
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Summary:Summary:
•The total Kinetic energy of a rotating system is The total Kinetic energy of a rotating system is the sum of the rotational energy about the Center the sum of the rotational energy about the Center of Mass and the translational KE of the CM. of Mass and the translational KE of the CM.
K = ½ IK = ½ ICMCM ⍵ ⍵ 22 + + ½ MR½ MR 2 2 ⍵ ⍵ 22
ττ = r x F = r x F
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
Summary:Summary:
Physics 1710Physics 1710—C—Chapter 11 Rotating hapter 11 Rotating BodiesBodies
•Angular momentum Angular momentum LL is the vector product of is the vector product of the moment arm and the linear momentum.the moment arm and the linear momentum.
L = r L = r x x pp
• The net externally applied torque is equal to The net externally applied torque is equal to the time rate of change in the angular the time rate of change in the angular
momentum.momentum.
∑ ∑ ττzz = d L = d Lzz /dt = I /dt = Izz ⍺ ⍺