Rotational Motion

1. Rotational Motion

2. Center of Gravity Center of Gravity (CG): an object’s average position of weight • for symmetrical objects the CG is at the geometric center of the object (ball) • the CG of a asymmetrical object is located closer to the end with the most mass (bat & toy) • Ex: the CG of a meterstick is at the 50 cm mark; the CG of a bat is towards the more massive end

3. Paths of CG • When objects are thrown, they will spin about their CG and move as if all the weight was concentrated about their CG. • The CG of thrown objects will follow are parabolic path. • Objects whose CG is not at their geometric center will “wobble” about the CG (bat)

4. When you slide an object across a horizontal table, the object will rotate about is CG while traveling in a straight line. • The white dot is the location of the wrench’s CG. Notice that it travels in a straight line

5. Locating the Center of Gravity • The CG of a uniform object (such as a meter stick) is at the midpoint, its geometric center • The CG of objects can be located where there is no matter Examples: A pot, chair, tea cup, boomerang

6. Check Your Understanding Where is the CG of a donut? • It’s in the center of the hole Can an object have more than one CG? • A rigid object has only one CG. If it is malleable, like putty or clay or the solar system, and is distorted into different shapes, then its CG may change as its shape is changed. Even then, there is only one CG for any given shape.

7. Toppling (Falling Over) • Rule for toppling: If the CG of an object is above its base of support, the object will remain upright. If the CG of an object extends beyond the base of support, the object will topple.

8. The Leaning Tower of Pisa • The Leaning Tower of Pisa does not topple because its CG is above its base of support

9. The base of support does not need to be solid. • The four legs of a chair bound a rectangular area that is the base of support for the chair

10. Check Your Understanding When you carry a heavy load – such as a pail of water – with one arm, why do you tend to hold your free arm out horizontally? • To shift your CG back over the support base

11. Check Your Understanding To resist being toppled, why does a wrestler stand with feet wide apart? • To increase the support base To resist being toppled, why does a wrestler stand with knees bent? • To lower the CG

12. Stability • An object is considered to be stable when work must be done to raise it’s CG • Three terms to describe an object’s stability • Stable equilibrium • Unstable equilibrium • Neutral equilibrium

13. Stable Equilibrium:any displacement raises an object’s center of gravity • Object is less likely to topple • Has a wider base of support • Ex: a pyramid Unstable Equilibrium:any displacement lowers an object’s center of gravity • Object is more likely to topple • Has a narrower base of support • Ex: an ice cream cone

14. Neutral Equilibrium:displacement neither raises nor lowers an object’s center of gravity • Object cannot topple over without putting work into it • Ex: a fish in water, a bat lying on the ground

15. The 1st cone is in unstable equilibrium, meaning it is more likely to topple • The 2nd cone is in stable equilibrium, meaning it is less likely to topple • The 3rd cone is in neutral equilibrium, meaning it can’t topple over anymore in that position

16. CG and Stability • An object becomes more stable when its CG is below the point of support. • Icebergs do not fall over because their CG are below the surface of the water • The CG of objects tend to take the lowest position available. • This is why sport players tend to squat lower to the ground, to stabilize themselves against an attack from an opponent.

17. Difference between torque and force • If you want to make an object move, apply a force • If you want to make an object rotate, apply a torque. • Forces produce acceleration. • Torque produce rotation

18. Torque Torque: the perpendicular force times the lever arm length • Torque produce rotations • The force must be perpendicular to the lever arm • Ex: turning a door knob; bending your arm • Equation: τ = F┴l • τ = Torque (N*m) • F┴ = Force perpendicular (N) • l = lever arm length (m)

19. The lever arm length is distance from the fulcrum to the area where the force is perpendicularly applied. Fulcrum: the pivot point of a lever • Where rotation begins • Ex: hinge of a door, center of a seesaw, your knee or elbow

20. Torque and lever arm length are directly proportional. • 2x lever arm length = 2 x Torque • ½ lever arm length = ½ Torque • 3x lever arm length = 3x Torque • 1/3 lever arm length = 1/3 Torque • Torque and force are directly proportional • 2x force = 2 x Torque • ½ force = ½ Torque • 3x force = 3x Torque • 1/3 force = 1/3 Torque

21. One way to produce more torque • Although the magnitudes of the applied forces are the same, the torques are different. Only the component of forces perpendicular to the lever arm contributes to the force.

22. Check Your Understanding If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench as shown?

23. No, because the lever arm (the arm of the wrench) is the same. If you wanted to increase the lever arm, you would need to use a longer wrench.

24. Balanced Torques • Weight does not produce rotation, torque does. • Consider a heavy boy and a slim boy on a seesaw. In order to balance the seesaw, the heavy boy must sit closer to the middle than the slim boy. In this way, the counter clock wise torque produced by the slim boy will equal the clock wise torque produced by the heavy boy.

25. Check Your Understanding Two children are on a seesaw. Child A is twice as heavy as Child B. Which one will sit closer towards the center to make them balanced? • Child A because a larger mass must have a smaller lever arm length to balance the smaller mass with the larger lever arm length of Child B.