PHYS16 – Lecture 27 Ch. 12 Static Equilibrium. Static Equil. Pre-question What is θ right before...

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PHYS16 – Lecture 27 Ch. 12 Static Equilibrium

Transcript of PHYS16 – Lecture 27 Ch. 12 Static Equilibrium. Static Equil. Pre-question What is θ right before...

PHYS16 – Lecture 27

Ch. 12 Static Equilibrium

Static Equil. Pre-question

• What is θ right before the box tips?

A) arctan(h/w)B) arctan(w/h)C) arctan(h/(2w))D) arctan(√(h2+w2)/2)E) None of the above.

F

FG

w

Static Equil. Pre-question

• Can an object be in equilibrium if it is in motion?

A) No, static equilibrium requires the object be at rest.B) No, unstable equilibrium requires that the derivative of the

potential be zero and the second derivative be less than zero.

C) Yes, mechanical equilibrium requires that the object have no acceleration.

D) Yes, static equilibrium requires that the sum of the forces and the sum of the torques equals zero.

E) None of the above.

Clarification

• Normal force for pole extending from wall

Ny

Nx

F

• Simple Machines – levers• Static Equilibrium– Center of Gravity– Newton’s Second law for torques and forces

• Stability Requirements

Ch. 12 Static Equilibrium

Stability

• Mathematically – Potential first deriv. = zero and second deriv. > zero and local min

• Stable equilibrium – if a small force acts to move the object from the equilibrium point and then disappears, the object will move back to the equilibrium point

• Ustable equilibrium – if a small force acts to move the object from the equilibrium point and then disappears, the object will move further from equilibrium

U

x

Where is object stable?

• Potential of an object is U=a(x4-2b2x2), where a and b are positive. Where is stable equilibrium?

yes then bwhen x

no then 0 when x,0412

b0, when x,0))((24422

23

abaxU

bxbxaxxabaxU

Practical Stability

• Practically – When center of mass is above the base of support

FGFG

Base of support

FG

Tipping point Unstable

FG

Stable

FG

Stable

Find Center of Mass

• Center of Mass = Center of Gravity if g = constant• Mathematically –

• Practically – assume g is constant, hang object by a support, center of mass is below point on straight line down from point

dVrrM

rCM )(1

Find Center of Mass

• Practically – need to hang by two different supports to uniquely locate center of mass

What is the center of mass of Massachusetts?

Discussion Question: Fancy wine rack

• Why does this wine rack work?• What if I add more bottles to the wine rack,

does it still work?• Why shouldn’t I buy this?

http://www.plunderhere.com/thumbnail.php?pic=uplimg/img_1859985_d6afcjx_btnwsogbgkkgrhqiokjyevllu3vbcbl8tr7vw0q_35.jpg&w=250&sq=N&b=N

Center of mass over base of support.

Works if added bottles create centerof mass over base of support.

If take bottle out -> fall

Discussion: Why build a pyramid?

http://images.goingtravels.com/2011/03/Giza-Pyramids.jpg

Base of support islarge

Incline gives mechanicaladvantage

Collapsing under weight

NN

F

Objects to Support Weight:BracketBraceButtressArch

Discussion: Skyscrapers

• Why were skyscrapers not built until 1880’s or so?

Elevator built 1850-1880’s

Static Equilibrium pre-question

• In an iron cross the gymnast is able to lift his weight with perfectly horizontal arms by:

A) Tensile forces – he pushes outwardB) Compressive forces – the rings pushes inwardC) Nonconservative forces – the gymnast uses his musclesD) None of the above. The iron cross does not require perfectly

horizontal arms.

Static Equil. Pre-question

• If ball 2 has twice the mass of ball 1 and the system is in static equilibrium then what is the ratio of L2 to L1?

A) 1:1B) 2:1C) 1:2D) There is not enough information.

L1L2

m2

m1

Static Equil. Pre-question

• What is θ right before the box tips?

A) arctan(h/w)B) arctan(w/h)C) arctan(h/(2w))D) arctan(√(h2+w2)/2)E) None of the above.

F

FG

w

Static Equil. pre-question

• Can an object be in equilibrium if it is in motion?A) No, static equilibrium requires the object be at rest.B) No, unstable equilibrium requires that the

derivative of the potential be zero and the second derivative be less than zero.

C) Yes, mechanical equilibrium requires that the object have no acceleration.

D) Yes, static equilibrium requires that the sum of the forces and the sum of the torques equals zero.

E) None of the above.

Conclusions

• Static Equilibrium – both forces and torques on an object have to sum to zero

• Practical Stability – center of gravity must be over base of support