Rotational Equilibrium

1. Rotational Equilibrium Chapter 11

2. Objectives • Describe how to make an object turn or rotate. • Explain what happens when balanced torques act on an object. • Describe how to find an object’s center of mass. • Describe how the center of gravity of an everyday object is related to its center of mass. • Describe how to predict whether an object will topple. • Explain why the center of gravity of a person is not located in a fixed place. • Describe what happens to the center of gravity of an object when the object is toppled.

3. 11.1 Torque Doorknob Hinge • When you open a door, you exert a force on the knob that turns the door about the hinge. • A force that produces rotational acceleration is called torque. Applying a torque makes an object turn or rotate.

4. 11.1 Torque • A torque is produced when a force is applied with “leverage.” • What this really means is that the force is applied to the end of a lever. • A force is applied to the end of this hammer (the lever) and the hammer claw rotates to pull the nail out. • In the previous example, the door is the lever.

5. 1.1 Torque • For a perpendicular force (F ), the distance from the turning axis to the point of contact of the force is called the lever arm. • The longer the lever arm, the greater the leverage and the easier the task. That means, with a longer lever arm, you can do the same job with less force. • For forces that are NOT perpendicular, only the perpendicular component of the force (F ) contributes to the torque. T T

6. 11.1 Torque • Torque is defined as: • Torque = Force applied • x lever arm • Or • t = F x d • (in this diagram d = r) • For the same torque, • you would need a • large force with a • short lever arm or a • small force with a long • lever arm. T

7. 1.1 Torque • As can be seen in the example above, the unit for torque is Newton-meter (N-m) when force is in N and d is in m. • Torque, like force, is a vector. It has a directional component.

8. Practice • You cannot exert enough torque to turn a bolt. What could you do? • If you exert 75 N on a wrench handle that is 0.3 m long, what torque are you exerting? • If you move your hand inward to be 0.2 m from the bolt, what force do you need to exert to have the same torque?

9. 11.2 Balanced Torques • A seesaw can be balanced even when weights are not equal. Weight alone does not produce rotation – torque does! • If the “heavier” children sit a shorter distance from the fulcrum, balance can be achieved because the torque producing a clockwise rotation equals the torque producing a counterclockwise rotation. • Fleft x d = Fright x d (Forces here equal the weights!) • When balanced torques act on an object, there is NO change in rotation.

10. ? Practice Figure 1 • What is the weight of the object on the left-hand side of the seesaw (Fig. 1)? • What is the distance from the pivot point to the 20 N mass in Fig. 2? • What is the mass of X in Fig. 3? 25 cm 20 N 35 N Figure 2 6.25 m 3.1 m 75 kg. X Figure 3

11. 11.3 Center of Mass • A ball thrown in to the air follows a smooth parabolic path. • The bat, however, seems to wobble all over the place. But, in truth, it wobbles about a special point that stays on a parabolic path.

12. 11.3 Center of Mass • The bat’s motion is the sum of 2 motions. • A spin around the “special” point • Movement through the air as if the mass were concentrated at this point. • This point, the center of mass, is where all the mass of an object is considered to be concentrated.

13. 11.3 Center of Mass • The center of mass of an object is the point located at the object’s average position of mass. • For symmetrical objects, the COM is at the geometric center. • For irregularly-shaped objects, the COM is towards the heavier end.

14. 11.3 Center of Mass • Objects that have a varying density throughout their structure may have a COM far from their geometric center. • These objects will “roll” and stop with their COM as LOW as possible.

15. Motion About theCenter of Mass • The COM of this gymnast follows a smooth parabolic path (like the bat in a previous slide). • The “parts” of the gymnast rotate around the COM. • The motion, therefore, is a combination of the motion of the COM and the rotation about the COM.

16. Motion About theCenter of Mass • The explosion of a projectile does not change the projectile’s COM. • The COM of the dispersed fragments will always be where the COM would have been had there been no explosion.

17. Applying Spinto an Object • When you apply a spin to an object, a force must be applied to the edge, AWAY from the COM. • This produces a torque that causes rotation.

18. 11.4 Center of Gravity • The COM, as you may recall, is defined as the average position of mass. • Center of gravity is defined as the average position of weight (the average position of all particles of weight that make up an object). • For everyday objects, they are interchangeable. • They are different for objects that are large enough to have varying degrees of gravity from one part to another. They are different for the Sears Tower, for example, because the lower floors are pulled more strongly by the Earth’s gravity than the upper ones.

19. Wobbling • The COG of the solar system lies outside the geometric center of the sun because the masses of the planets contribute to the overall mass of the solar system. • This causes the sun to wobble. Astronomers look for stars that wobble, as it indicates the star has a planetary system.

20. Locating the Centerof Gravity • The COG of a uniform object will be at its geometric center. • The small force of gravity vectors of the object can be combined into a single resultant force at the COG. (Fig. 11.13) Its as if the weight of the object is concentrated at this point. • It is the balance point – supporting that point supports the whole object.

21. Location of the Center of Gravity A simple way to find the COG is illustrated above.

22. Location of the Center of Gravity • The COG of an object may be located where no actual mass exists. • The COG of a ring (or even ½ a ring) lies in the center. • The COG of a hollow sphere (or even ½ a sphere) lies in the center.

23. Practice • Where must a soccer ball be kicked if you wish it to spin as it moves through the air? • Where is the COG of an empty cup? Of a donut? • Can a ball of clay have more than one COG?

24. 11.5 Torque and Centerof Gravity • The rule for toppling: if the COG of an object is ABOVE the area of support, the object will remain upright. • The Leaning Tower of Pisa does not topple over because its COG lies above its base.

25. 11.5 Torque and Centerof Gravity • A plumb line shows that, in the last diagram, the vehicle will topple. • This is because the plumb line extends beyond the supporting base of the vehicle.

26. 11.5 Torque and Centerof Gravity • If the COG extends outside the area of support, an unbalanced torque exists. • An unbalanced torque will topple an object. • Note: the support base does not have to be solid. Refer again to the vehicles in the previous slide. The tires create a rectangular area that is the support base for the vehicle.

27. Practice • Which truck(s) will topple? • To resist being toppled, why does a wrestler stand with: • His feet wide apart? • He knees bent?

28. Balancing • In learning to walk, a baby must lean to coordinate and position her COG above a supporting foot. • Eventually, her brain will be able to coordinate the adjustments for maintaining balance quite quickly and easily.

29. The Moon’s COG • Because the side of the moon facing Earth has a greater gravitational pull on it from the Earth than the moon’s farther parts, the moon’s COG is closer to Earth than its COM. • The moon rotates about its COM and the Earth pulls on its COG. • When the moon’s COG is NOT lined up between the Earth’s and moon’s center, a torque is produced that keeps one side of the moon continually facing Earth.

30. 11.6 Center of Gravityof People • The COG of a person is NOT located in a fixed place. It depends on body orientation. • When standing, your COG is normally 2-3 cm below your navel, halfway between your front and back. • The COG of women is slightly lower because they have smaller shoulder and a larger pelvis. • The COG of children is higher because their heads tend to be larger and their legs shorter.

31. 11.6 Center of Gravityof People • Once you move, your COG changes. • If you bend your body into a C shape, the COG is located outside the body!

32. 11.6 Center of Gravityof People • High jumpers use physics in a move called the “Fosbury flop.” • The high jumper can clear the bar while his COG passes beneath the bar.

33. 11.6 Center of Gravityof People • Your support area is generally the area bounded by your feet. • You can increase this area by placing your feet further apart or by leaning on another object.

34. Why or Why Not? • Sit in a chair. Try to stand up WITHOUT putting your feet under the chair. • Stand facing a wall with toes against the wall. Now stand on tip toes for a few seconds. • Bend over and touch your toes without bending your knees. Repeat, this time with your heels against a wall. • Carry a heavy pail with one arm. Keep the other arm against your side.

35. 11.7 Stability • It is rather easy for the cone to stand upright on its flat end. • We say it is in stable equilibrium. It is balancedso thatany change in its position will RAISE its COM.

36. 11.7 Stability • If we take that cone and do work on it, we can increase its potential energy by raising its COG. • Even though the COG is exactly above the tip of the cone, it is in unstable equilibrium. Any movement or change in position of the cone will LOWER its COG.

37. 11.7 Stability • If the cone then topples over, any small movement to it will neither raise nor lower the COG. • It is in neutral equilibrium.

38. 11.7 Stability • Consider a book lying flat and one that is upright. Both are in stable equilibrium. Why? • The flat book is more stable, though. It will take MORE work to raise the COG of the flat book than the upright one. • An object with a low COG is usually more stable than one with a high COG.

39. Objects in Stable Equilibrium • The plank balanced in Fig. A is in unstable equilibrium – its COG is lowered if it tilts. • But, by adding 2 masses to the plank, as in Fig. B, it becomes stable. If the plank is tilted, the COG is raised. Figure A Figure B

40. Objects in Stable Equilibrium • “Balancing toys” depend on this principle. • Their COG is below the point of support. The COG will rise when it tilts so the toy is in stable equilibrium.

41. Objects in Stable Equilibrium • The Space Needle in Seattle is in stable equilibrium. • Much of its structure is below ground so that its COM is also below ground. • It cannot fall over intact because such an action would actually raise its COG.

42. Lowering the COG of an Object • The COG of an object tends to take the lowest position available. • This is because the heavier (more dense) material of an object tends to occupy the available lower space.

43. Lowering the COG of an Object • It’s like “panning for gold.” • Material of greater density (like gold) takes up position in the bottom of the pan. This effectively lowers the COG of the material in the pan.