1. Circular Motion, Center of Gravity, & Rotational Mechanics
Chapters 9, 10, & 11

2. Rotation and Revolution
Axis – straight line around which rotation occurs
Rotation – object turns about an internal axis (Earth rotates around its axis)
Revolution – when an object turns about an external axis (Earth revolves around the sun)

3. Rotation

4. Revolution

5. Rotational Speed
The linear speed of an object is greater near the outer edge of a rotating object than on the inner edge of the rotating object
Tangential Speed – the speed of something moving along a circular path (the direction of motion is always tangent to the circle)
Rotational Speed ( angular speed) – the number of rotations per unit of time (expressed in revolutions per minute or RPM)
All parts of a rotating object rotate about their axis in the same amount of time!

6. Uniform Circular Motion

7. Tangential Speed ~ Radial Distance x Rotational Speed
Tangential speed and rotational speed are related
Tangential Speed ~ Radial Distance x Rotational Speed
As you move away from the center of a rotating object, the tangential speed will increase while your rotational speed stays the same

8. Centripetal Force
Centripetal Force – any force that causes an object to follow a circular path
When a car goes around a corner, the friction between the tires and the road provides the centripetal force needed to keep the car going around the curve
If not for the friction of the tires, the car would continue moving in the straight-line path

9. Centripetal Force

10. To Have or Not to Have Centripetal Force
Without Centripetal Force
With Centripetal Force

11. Centrifugal Force
Centrifugal Force – the outward force associated with circular motion
It is not a true force, but rather the effect that inertia tries to place on you as you follow a circular path
From Newton’s First Law, the natural path of an object is a straight-line, the centripetal force is what keeps you going in a circle

12. Centrifugal Force

13. Center of Gravity
Center of Gravity – point located at the object’s average position of weight
For a symmetrical object, it is the geometric center of the object
For an irregularly shaped object, there is more weight on end than the other, so the center of gravity is toward the heavier end
Objects not made of the same material throughout (different densities), may have the center of gravity very far from the geometric center

14. Center of Gravity

15. Center of Mass
Center of Mass – the average position of all the particles of mass that make up an object
For almost all objects on or near Earth, center of gravity and center of mass are interchangeable
If you threw an object in the air, you’d see it wobble around its center of gravity
The sun wobbles also!
As the planets move around the sun, they contribute to the overall center of mass of the solar system, so the sun wobbles off center
This is how astronomers look for planets orbiting other stars!

16. Center of Mass in a Star System

17. Locating the Center of Gravity
The center of gravity is the balance point, supporting that single point supports the whole object
If you suspend any object at a single point, the center of gravity for that object will hang directly below (or at) the point of suspension
The center of gravity may be located where no actual material exists (i.e. a ring)

18. Locating the Center of Gravity

19. Toppling
If the CG of an object is above the area of support, it will remain upright
If the CG extends outside the area of support, the object will topple
The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base

20. Toppling

21. Stability
Unstable Equilibrium – an object balanced so that any displacement lowers its center of gravity
Stable Equilibrium – an object balanced so that any displacement raises its center of gravity (requires work)
Neutral Equilibrium – an object balanced so that its center of gravity is neither raised nor lowered with displacement

22. Stability

23. Center of Gravity of People
When you stand upright with your arms hanging at your sides, your CG is within your body, typically 2 to 3 cm below your belly button
The CG is slightly lower in women than in men, because women tend to be proportionally larger in the pelvis and smaller in the shoulders
When you stand, your CG is somewhere above the support base of your feet, we spread them further apart in unstable situations (the bus)
When you bend over to touch your toes, you are unconsciously extending the lower part of your body, putting your CG outside of your body (so you won’t topple over!)

24. Center of Gravity of People

25. Torque = force┴ x lever arm
A torque is produced when a force is applied with “leverage”
You use leverage when you use a screwdriver to open a can of paint
The direction of your applied force is important, you would never try to open a door with a doorknob by push or pulling the doorknob sideways
You apply your force PERPENDICULAR to the plane of the door
Torque = force┴ x lever arm
Greater torques are produced when both the force and lever arm are large

26. Torque

27. Torque and Center of Gravity
If the direction of force is through the CG of the projectile, all the force can do is move the object as a whole; there will be no torque to turn the projectile
If the force is directed “off center”, then in addition to motion of the CG, the projectile will rotate about its CG

28. Rotational Inertia
An object rotating about an axis tends to keep rotating about that axis (look familiar?)
Rotational Inertia – the resistance of an object to changes in its rotational motion
A torque is required to change the rotational state of motion of an object
Rotational inertia depends on the distribution of the mass of an object

29. Rotational Inertia

30. Rotational Inertia and Gymnastics
Rotational inertia is least about the vertical head-to-toe axis (longitudinal) on any person, because most of the mass is concentrated there
A rotation of your body along this axis is easiest
The rotational inertia when your arms are extended is 3 times greater than when your arms are pulled in
You rotate about your transverse axis when you do a flip or somersault
The rotational inertia of a gymnast is up to 20 times greater when she is swinging in a fully extended position from a horizontal bar than after dismount when she somersaults in a tucked position (when she let goes and tucks, she is automatically increasing her rate of rotation by 20 times!)

31. Rotational Inertia and Gymnastics

32. Angular Momentum
Angular Momentum – the “inertia of rotation” of rotating objects
Like linear momentum, angular momentum is a vector quantity
Rotational Velocity – when a direction is assigned to rotational speed
Angular momentum = rotational inertia (I) x rotational velocity (ω)
Angular momentum = mass (m) x velocity (v) x radius (r)
Newton’s 1st Law can now be restated for angular momentum:
An object or system of objects will maintain its angular momentum unless acted upon by an unbalanced external torque

33. Angular Momentum

34. Conservation of Angular Momentum
The Law of Conservation of Angular Momentum:
If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant.
With no external torque, the product of rotational inertia and rotational velocity at one time will be the same at any other time

35. Conservation of Angular Momentum

36. Assignment Read Ch. 9-11 (pg. 122-164)
Ch. 9: Do #31-38 (pg. 135), Appendix F #1-7 (pg. 674)
Ch. 10: Do #21-34 (pg )
Ch. 11: Do #33-40 (pg. 167)