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PHYS16 – Lecture 27

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 (√(h 2 +w 2 )/2) E) None of the above. w. F. h. θ. F G. Static Equil . Pre-question.

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PHYS16 – Lecture 27

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  1. PHYS16 – Lecture 27 Ch. 12 Static Equilibrium

  2. 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. w F h θ FG

  3. Static Equil. Pre-question • Can an object be in equilibrium if it is in motion? • No, static equilibrium requires the object be at rest. • No, unstable equilibrium requires that the derivative of the potential be zero and the second derivative be less than zero. • Yes, mechanical equilibrium requires that the object have no acceleration. • Yes, static equilibrium requires that the sum of the forces and the sum of the torques equals zero. • None of the above.

  4. Clarification • Normal force for pole extending from wall F Ny Nx

  5. Ch. 12 Static Equilibrium • Simple Machines – levers • Static Equilibrium • Center of Gravity • Newton’s Second law for torques and forces • Stability Requirements

  6. 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

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

  8. Practical Stability • Practically – When center of mass is above the base of support FG FG FG FG FG Stable Stable Tipping point Unstable Base of support

  9. 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

  10. 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?

  11. 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? Center of mass over base of support. Works if added bottles create center of mass over base of support. If take bottle out -> fall http://www.plunderhere.com/thumbnail.php?pic=uplimg/img_1859985_d6afcjx_btnwsogbgkkgrhqiokjyevllu3vbcbl8tr7vw0q_35.jpg&w=250&sq=N&b=N

  12. Discussion: Why build a pyramid? Base of support is large Incline gives mechanical advantage http://images.goingtravels.com/2011/03/Giza-Pyramids.jpg

  13. Collapsing under weight N N F Objects to Support Weight: Bracket Brace Buttress Arch

  14. Discussion: Skyscrapers • Why were skyscrapers not built until 1880’s or so? Elevator built 1850-1880’s

  15. Static Equilibrium pre-question • In an iron cross the gymnast is able to lift his weight with perfectly horizontal arms by: • Tensile forces – he pushes outward • Compressive forces – the rings pushes inward • Nonconservative forces – the gymnast uses his muscles • None of the above. The iron cross does not require perfectly horizontal arms.

  16. 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) 1:1 B) 2:1 C) 1:2 D) There is not enough information. L2 L1 m1 m2

  17. 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. w F h θ FG

  18. Static Equil. pre-question • Can an object be in equilibrium if it is in motion? • No, static equilibrium requires the object be at rest. • No, unstable equilibrium requires that the derivative of the potential be zero and the second derivative be less than zero. • Yes, mechanical equilibrium requires that the object have no acceleration. • Yes, static equilibrium requires that the sum of the forces and the sum of the torques equals zero. • None of the above.

  19. 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

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