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(√(h 2 +w 2 )/2) E) None of the above. h θ F FGFG w

3. 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.

4. Clarification Normal force for pole extending from wall NyNy NxNx F

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

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(x 4 -2b 2 x 2 ), where a and b are positive. Where is stable equilibrium?

8. Practical Stability Practically – When center of mass is above the base of support FGFG FGFG Base of support FGFG Tipping pointUnstable FGFG Stable FGFG

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? 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 center of mass over base of support. If take bottle out -> fall

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

13. Collapsing under weight NN 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: A)Tensile forces – he pushes outward B)Compressive forces – the rings pushes inward C)Nonconservative forces – the gymnast uses his muscles D)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 L 2 to L 1 ? A) 1:1 B) 2:1 C) 1:2 D) There is not enough information. L1L1 L2L2 m2m2 m1m1

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(√(h 2 +w 2 )/2) E) None of the above. h θ F FGFG w

18. 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.

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