physics 1710—warm-up quiz why does a diver rotate faster when she tucks in her arms and legs?...

Physics 1710Physics 1710—Warm-up Quiz—Warm-up QuizWhy does a diver rotate faster when she tucks Why does a diver rotate faster when she tucks in her arms and legs?in her arms and legs?

A B C D E

6%

24%

2%4%

65%

A.A. She increases her angular momentum.She increases her angular momentum.

B.B. She increases her moment of inertia.She increases her moment of inertia.

C.C. She decreases her moment of inertia.She decreases her moment of inertia.

D.D. She pushes against her inertia.She pushes against her inertia.

E.E. None of the aboveNone of the above

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Analysis:Analysis:

• Like an ice skater. Why does an ice skater Like an ice skater. Why does an ice skater increase her angular velocity without the benefit increase her angular velocity without the benefit of a torque?of a torque?

L = r x p= r x L = r x p= r x ( m ( m vv))= = r x r x ( m ( m r x r x ⍵⍵))

LLii = m = mii r rii 2 2 ⍵ ⍵ zz

LLzz = (∑ = (∑ii m mii r rii 2 2 ) ⍵ ) ⍵zz

LLzz = I = Izz ⍵ ⍵z z ; & ⍵; & ⍵zz = L = Lzz / I / Izz

Therefore, a decrease in I ( by reducing r) will Therefore, a decrease in I ( by reducing r) will result in an increase in ⍵ even if dL/dt = 0!result in an increase in ⍵ even if dL/dt = 0!

Physics 1710Physics 1710—C—Chapter 13 App: E & Ehapter 13 App: E & E

Rotating Platform Rotating Platform DDemonstrationemonstration


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 ⍵.


No Talking!No Talking!

Think!Think!

Confer!Confer!

Peer Instruction Peer Instruction TimeTime

How does a ladder stay up?How does a ladder stay up?


11′′ Lecture Lecture •Static equilibriumStatic equilibrium (no translational or rotational (no translational or rotational acceleration) requires that all forces and torques to acceleration) requires that all forces and torques to balancebalance..

• If the acceleration due to gravity is the same for If the acceleration due to gravity is the same for all particles comprising a body the center of mass all particles comprising a body the center of mass is the center of gravity.is the center of gravity.

CM = CGCM = CG if g the same for all m if g the same for all mi i . .

• The moduli of elasticity (Y, E, B) characterizes The moduli of elasticity (Y, E, B) characterizes the stress-strain relation:the stress-strain relation:

• stress= modulus • strainstress= modulus • strain; ; σ = Y εσ = Y ε


One more turn on One more turn on L—L—Angular Momentum:Angular Momentum:


L = r x pL = r x p

The angular momentum is the The angular momentum is the vector product vector product of the moment arm and the linear momentum.of the moment arm and the linear 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.

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


Second Law of MotionSecond Law of Motion

L = L = constant meansconstant means angular momentum is conserved.


If If ττ = 0, then = 0, then

LL is a constant. is a constant.


II ω

Second Law of Motion & Gyroscopic Second Law of Motion & Gyroscopic PrecessionPrecession

ττ = = r x F r x F = = d d LL//dtdt

= = II (d(dω/dt); ddω/dt = II-1-1 ττ


= d( = d( II ω)/dt

L = II ω

F = mg

rττ


No Talking!No Talking!Think!Think!

Confer!Confer!


What will happen to a tilted spinning top when it is supported on one end only? Why?


What will happen to a tilted, spinning top when it is supported on one end only? Why?

A B C D E

0%

11%

5%

54%

30%

A.A. It will fall over because of gravity. It will fall over because of gravity.

B.B. It will spin faster because of I is It will spin faster because of I is changing.changing.

C.C. It will precess clockwise due to It will precess clockwise due to torque.torque.

D.D. It will precess counterclockwise It will precess counterclockwise due to torque.due to torque.

E.E. None of the above.None of the above.

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Experiment:Experiment:

Spinning TopSpinning Top

•Observe the direction of rotationObserve the direction of rotation ( (ω)

• Observe the motion (dObserve the motion (dω/dt)/dt)


II ω

Second Law of Motion & Gyroscopic Second Law of Motion & Gyroscopic PrecessionPrecession

ττ = = r x F r x F = = d d LL//dtdt

= = II (d(dω/dt); ddω/dt = II-1-1 ττ


= d( = d( II ω)/dt

L = II ω

F = mg

rττ


Torque and the Right Hand Rule:Torque and the Right Hand Rule:

rr

FF r r x x FF

X


II ωGyroscopic PrecessionGyroscopic Precession


= d( = d( II ω)/dt

= = II (( ddω/dt)

(Top view)

ττ


Fundamental Angular MomentumFundamental Angular Momentum

Fundamental unit of angular momentum = Fundamental unit of angular momentum = ℏℏℏℏ = 1.054 x 10 = 1.054 x 10 -34-34 kg‧m/s kg‧m/s22

IICMCM⍵ ≈ ⍵ ≈ ℏℏ⍵⍵ ≈ ≈ ℏ ℏ / / IICMCM

= = 1.054 x 10 1.054 x 10 -34-34 kg‧m/s kg‧m/s22 / (1.95 x 10 / (1.95 x 10 -46-46 kg‧m) kg‧m)= 5.41 x 10= 5.41 x 10 11 11 rad/s rad/s


MRI (a.k.a. NMR: Nuclear Magnetic MRI (a.k.a. NMR: Nuclear Magnetic Resonance)Resonance)

MMagnetic agnetic RResonance esonance IImagingmaging


Hydrogen atoms

Magnetic Field Torque=?

Precession!

MRI permits imaging of soft tissue due precession of the hydrogen in the water of the human body.

Teeter-totter: How does it Teeter-totter: How does it balance?balance?

ττnet net == ΣΣττii

FF22

FF11


FFgg==FF11++FF22

FFsupport support = - = - FFgg

For equilibrium:For equilibrium:

ττnet net == ΣΣττii == 0 0

FFnet net == ΣΣFFii == 0 0


Good ideas?

How does a ladder stay up?How does a ladder stay up?

Legends of the Fall—How a ladder stays upLegends of the Fall—How a ladder stays up

FFnetnet = = 00

FFnetnet = = 00

TTnetnet = = 00

Physics 1710Physics 1710—C—Chapter 11 App: E hapter 11 App: E & E& E

Equilibrium: Equilibrium: (equi= equal (equi= equal “=“;“=“; libium = scales libium = scales “ “ ♎”♎”))

Equilibrium implies “balanced.”Equilibrium implies “balanced.”

FFnetnet = = d d PP/dt/dt

In equilibrium In equilibrium FFnetnet = = 00

ττnetnet = = d d LL/dt/dt

In equilibrium In equilibrium ττnetnet = = 00

Physics 1710Physics 1710—C—Chapter 11 App: E hapter 11 App: E & E& E

Can an object be in equilibrium when the center of mass lies outside of the object?

A B C

56%

26%

18%

A.A. Yes. Yes.

B.B. No.No.

C.C. Depends.Depends.

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Solution:Solution:


Hollow Cone

In equilibrium In equilibrium

FFnetnet = = 00

ττnetnet = = 00

Is it “stable?”

i.e. does it recover from a small

displacement ?

Center of GravityCenter of Gravity

If the acceleration due to gravity is the same for If the acceleration due to gravity is the same for all parts of a body, then the center of mass all parts of a body, then the center of mass corresponds to the center of gravity. corresponds to the center of gravity.

CG =CM if g uniformCG =CM if g uniform

Proof: TProof: Ti i = = ΣxΣxi i FFgigi = Σx = Σxi i gmgmgigi = Mg Σx = Mg Σxi i mmgigi /M= F /M= Fcgcg xxcmcm


Center of Gravity (comparison)Center of Gravity (comparison)


BallBallFFg g = - mg= - mg

MoonMoonFFg g = - GmM/r= - GmM/r22

CM

CMCG

ElasticityElasticityDefinitions:Definitions:

• Stress Stress σσ : the deforming force per unit : the deforming force per unit area.area.

• Strain Strain εε : the unit deformation. : the unit deformation.

Stress = modulus x strainStress = modulus x strainσ = σ = F/A = Y F/A = Y εε


ElasticityElasticity

• Stress Stress σσ – Strain – Strain εε “Curve” “Curve”


Strain Strain ε = ΔL/L (%)ε = ΔL/L (%)

Str

ess

S

tress

σ

σ

(N

/m(N

/m22))

σ = σ = Y Y εε

Elastic limit

Failure


• Stress Stress σ σ : the deforming force per unit : the deforming force per unit area.area.

• Strain Strain ε ε : the unit deformation.: the unit deformation.

Tensile/Compressive Stress Tensile/Compressive Stress Young’s ModulusYoung’s Modulus EE

Stress = modulus x strainStress = modulus x strainσ = σ = F/A = E F/A = E ε = E ε = E ΔL/LΔL/L

σ σ = E = E ε ε ΔLΔLLL





Shear Modulus GShear Modulus GStress = modulus x strainStress = modulus x strain

σ = σ = F/A = G F/A = G ε = G ε = G Δx/LΔx/L

σσΔxΔxLL





Hydraulic Stress: Hydraulic Stress: Bulk Modulus BBulk Modulus B

Stress = modulus x strainStress = modulus x strainσ = σ = F/A = p = B F/A = p = B ε = B ε = B ΔV/VΔV/V

pp


VV

ΔVΔV


• Stress: the deforming force per unit Stress: the deforming force per unit area.area.

• Strain: the unit deformation.Strain: the unit deformation.

– Tensile: “stretch”Tensile: “stretch”– Compressive: “squeeze”Compressive: “squeeze”– Shear: “lean”Shear: “lean”– Hydraulic: pressureHydraulic: pressure– Yield: permanently deformedYield: permanently deformed


SummarySummary

• Static equilibrium implies that all forces Static equilibrium implies that all forces and torques balance.and torques balance.

• The center of mass is often the center of The center of mass is often the center of gravity.gravity.

• The moduli of elasticity characterizes the The moduli of elasticity characterizes the stress-strain relation:stress-strain relation:

• stress= modulus x strainstress= modulus x strainStress = modulus x strainStress = modulus x strain

σ = σ = F/A = Y F/A = Y εε


No Talking!No Talking!Think!Think!

Confer!Confer!


Why does the platform spin faster Why does the platform spin faster when he brings his arms in?when he brings his arms in?


Why does the platform spin faster when he Why does the platform spin faster when he brings his arms in?brings his arms in?

1010

A B C D E

0% 0% 0%0%0%

A.A. He increases his angular He increases his angular momentum.momentum.

B.B. He increase his moment of He increase his moment of inertia.inertia.

C.C. He decrease his moment of He decrease his moment of inertia.inertia.

D.D. He pushes against the He pushes against the inertia of the weights.inertia of the weights.

E.E. None of the aboveNone of the above

Answer Answer Now !Now !

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Where should the fulcrum be place to balance the teeter-totter?

1010

A B C

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A.A.

B.B.

C.C.


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Which way will the torque ladder move?1010

A B C

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A.A. Clockwise Clockwise B.B. CounterclockwiseCounterclockwiseC.C. Will stay balancedWill stay balanced


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physics 1710—warm-up quiz why does a diver rotate faster when she tucks in her arms and legs?...

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