Center of Mass and Universal Gravitation

1. Center of Mass and Universal Gravitation Center of Mass material is informational: you need to know the concept to understand gravitation

2. Why doesn’t the Leaning of Tower of Pisa topple over?

3. How far can it lean before it does topple over?

4. To answer these questions we first need to know about Center of Mass

5. Throw a baseball into the air and it follows a smooth parabolic path. Throw a baseball bat and the bat seems to wobble all over the place path of object throwing object

6. Throwing a Baseball Bat • It wobbles about a special point. • This point stays on the parabolic path, even though the rest of the bat does not • The motion of the bat is the sum of: • A spin around this point • A movement through the air as if the mass were concentrated at this point • This point is called the center of mass. Path of the other end Path of the handle General Path

7. Center of Mass • What it is not • The halfway point of the mass • The arithmetic average of the mass • What it is • The place where mass x distance for each piece “balances out” • The point where all our earlier (non-rotation) motion laws apply

8. Gravity up here… … is about .014% more than down here Center of Gravity • Almost identical to CM, but different for extremely large objects • For very tall buildings, the earth’s gravitation field differs between the top and bottom floors • CM could be different from CG by a few centimeters

9. Location of CM • For a completely symmetrical object, such as a baseball, the center of mass is at the geometric center of the object. • For an object like a bat, the center of mass is toward the heavier end. • Objects not made of the same material throughout may have the center of mass quite far from the geometric center.

10. How to Find the CG • Balance the object two different ways • CM must be in line above balance point • Intersection of 2 lines-- CM • Suspend the object two different ways • CM must be in line under suspension point • Intersection of 2 lines-- CM

11. CM of People • ..\HighJump-0001.mpeg • ..\high jump.mpeg

12. Toppling • Why do objects fall over?

13. Stability • An object will not topple (fall over) if its CM is above its area of support.

14. Examples of Center of Mass • Spoon and fork • Saw and ruler • “tightwire bicycle”

15. Universal Gravitation • Newton came up with the “Law of Universal Gravitation”: • objects with mass have a gravitational attraction toward each other F = G m1 m2 r2 • r is the distance between their centers of mass-- NOT their surfaces • G is universal gravitational constant: • 6.67 x 10 -11 Nm2/kg2 m1 F r F m2

16. Example • What’s the force of gravity on a 100 kg astronaut at the space station 0.39 x106 m above Earth? • Earth’s radius = 6.37x106 m, so space sta. is 6.76x106 m from center of earth • Earth’s mass = 6 x 1024 kg • G = 6.67 x 10 -11 Nm2/kg2 F = G m1 m2=(6.67 x 10 -11)(100)(6 x 1024 ) r2 (6.76x106 )2 = 876 N Does this seem big?

17. What does it mean? F = G m1 m2 r2 • If I double the mass of one of the objects, I double the gravitational force m1 F r F m2

18. What does it mean? F = G m1 m2 r2 • As the distance increases, the force goes way down • “The inverse square law” • Twice as far apart, ¼ the force; 10 times as far apart, 1/100 of the force • Make sure you use the distance between the centers of mass: for a person standing on the Earth’s surface, the distance is the radius of the Earth!

19. Solving Gravitational Problems F = G m1 m2 r2 • If they give you the numbers, plug ‘em in • If they just say something like “four times as much mass”: • Force vs. mass is linear, so there will be 4 times as much gravitational force • If they say something like “four times the distance”: • Force vs. distance is inverse-squared, so you divide by 42 • There will be 1/16 as much gravitational force

20. Inverse Square Law • Think of a square beam of light • At some distance it covers a 1 foot x 1 foot square • Twice as far away, it covers a 2 foot x 2 foot square: 4 times the area • But ¼ the brightness– the same light is spread over a broader area • Gravitational force seems to work in a similar way

21. Orbits

22. What You Need To Be Able To Do • Know and apply the gravitation equation • Address parametric changes • E.g., “if I triple the distance, how does the gravitational force change?” • Explain that an object in a circular orbit travels with a constant linear speed and is kept in orbit by the gravitational force acting centripetally • Associate lower orbits with faster velocities