1. Equilibrium Lecturer: Professor Stephen T. Thornton

2. Reading Quiz A) all B) 1 only C) 2 only D) 3 only E) 2 and 3 A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dot in each case. In which case(s) does the box tip over? 123

3. Reading Quiz 123 A) all B) 1 only C) 2 only D) 3 only E) 2 and 3 A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dot in each case. In which case(s) does the box tip over? If the box can rotate such that the CM is lowered, it will!! The torque due to gravity acts like all the mass of an object is concentrated at the CM. Consider the bottom right corner of the box to be a pivot point. If the box can rotate such that the CM is lowered, it will!!

4. Last Time Angular momentum and torque Gyroscopes Inertial forces, pseudoforces Review

5. Today Force equilibrium Rotational equilibrium Stability and balance Elasticity; stress and strain

6. Copyright © 2009 Pearson Education, Inc. The Conditions for (force) Equilibrium

7. Copyright © 2009 Pearson Education, Inc. The second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary. The Conditions for (torque) Equilibrium

8. Forces Required for Equilibrium

9. Is the plank in equilibrium? The forces may be balanced, but is the torque balanced? Maybe not. Condition for force equilibrium: Look carefully at plank problem. Make sure you understand how to get the torques. Find torque about any pivot. Condition for rotational equilibrium:

10. A Lamp in Equilibrium

11. Do rotational equilibrium demo.

12. Zero Torque and Balance This is primarily a torque problem. Why is that?

13. Zero Torque and Balance This is primarily a torque problem. Why is that?

14. The Geometric Center of the United States Do CM of Virginia demo again.

15. Conceptual Quiz: A box, with its CM off center as indicated by the dot, is placed on an inclined plane. In which of the four orientations shown, if any, does the box tip over? A) A B) B C) C D) D E) Box does not tip over. A D C B

16. Answer: C This is the only orientation where the CM is to the left of the likely pivot point on the bottom left corner of the box.

17. Conceptual Quiz 1kg 1m A 1-kg ball is hung at the end of a rod 1-m long. If the system balances at a point on the rod 0.25 m from the end holding the mass, what is the mass of the rod? A) kg B) kg C) 1 kg D) 2 kg E) 4 kg

18. 1 kg X CM of rod same distance m ROD = 1 kg A 1-kg ball is hung at the end of a rod 1-m long. If the system balances at a point on the rod 0.25 m from the end holding the mass, what is the mass of the rod? The total torque about the pivot must be zero !! 0.25 m to the right of the pivot same distance to the left of the pivot The total torque about the pivot must be zero !! The CM of the rod is at its center, 0.25 m to the right of the pivot. Because this must balance the ball, which is the same distance to the left of the pivot, the masses must be the same !! A) kg B) kg C) 1 kg D) 2 kg E) 4 kg Conceptual Quiz

19. Answer: C 0.25 m of stick on each side cancels out. Next 0.5 m of stick must balance rock. Distance is 2X, mass in second half of stick must be ½ kg. Total mass of stick is then 1 kg.

20. 1.Choose one object at a time, and make a free- body diagram by showing all the forces on it and where they act. 2.Choose a coordinate system and resolve forces into components. 3.Write equilibrium equations for the forces. 4.Choose any axis perpendicular to the plane of the forces and write the torque equilibrium equation. A clever choice here can simplify the problem enormously. 5.Solve. Solving Statics Problems

21. If a force in your solution comes out negative (as A will here), it just means that it’s in the opposite direction from the one you chose. This is trivial to fix, so don’t worry about getting all the signs of the forces right before you start solving. Solving Statics Problems

22. Propped Board. A large 62.0-kg board is propped at a 45° angle against the edge of a barn door that is 2.6 m wide. How great a horizontal force must a person behind the door exert (at the edge) in order to open it? Assume that there is negligible friction between the door and the board but that the board is firmly set against the ground.

23. If the forces on an object are such that they tend to return it to its equilibrium position, it is said to be in stable equilibrium. Stability and Balance

24. If, however, the forces tend to move it away from its equilibrium point, it is said to be in unstable equilibrium. Stability and Balance

25. Copyright © 2009 Pearson Education, Inc. An object in stable equilibrium may become unstable if it is tipped so that its center of gravity is outside the pivot point. Of course, it will be stable again once it lands! Stability and Balance

26. Copyright © 2009 Pearson Education, Inc. People carrying heavy loads automatically adjust their posture so their center of mass is over their feet. This can lead to injury if the contortion is too great. Stability and Balance

27. Demos: CM toys

28. Do demo of blocks on edge of table.

29. Conceptual Quiz 1 kg 1 m3 m 1 m2 m ? ? A (static) mobile hangs as shown below. The rods are massless and have lengths as indicated. The mass of the ball at the bottom right is 1 kg. What is the total mass of the mobile? A) 5 kg B) 6 kg C) 7 kg D) 8 kg E) 9 kg

30. 1 kg 1 m3 m 1 m2 m ? ? A (static) mobile hangs as shown below. The rods are massless and have lengths as indicated. The mass of the ball at the bottom right is 1 kg. What is the total mass of the mobile? Finally, add up all the masses. Use torques in two steps: (1) find the big mass on the bottom left (lower rod only), and (2) use the entire lower rod assembly (with two masses) to find the mass on top right. Finally, add up all the masses. Conceptual Quiz A) 5 kg B) 6 kg C) 7 kg D) 8 kg E) 9 kg

31. Conceptual Quiz A) case 1 will tip B) case 2 will tip C) both will tip D) neither will tip Consider the two configurations of books shown below. Which of the following is true? 1/2 1/4 1/2 1/4 12

32. Conceptual Quiz CM of case #1 is not over the table, so it will tip The CM of the system is midway between the CM of each book. Therefore, the CM of case #1 is not over the table, so it will tip. A) case 1 will tip B) case 2 will tip C) both will tip D) neither will tip Consider the two configurations of books shown below. Which of the following is true? 1/2 1/4 1/2 1/4 12