1. Project

2. Lesson #65 Topic: Center of Mass / Gravity Objectives: (After this class I will be able to) 1. Define center of gravity / mass 2. Locate an object’s center of mass 4/18/07 Warm Up: Toss different objects through the air and observe differences and similarities of their motion. Assignment: Book Problems p147 RQ #5, 7, 8, 11,12

3. Center of gravity / mass Any object can be treated as a particle with all of its mass concentrated at a single point. Any object can be treated as a particle with all of its mass concentrated at a single point. This point is called the objects Center of Mass. This point is called the objects Center of Mass. The point where the force of gravity (weight) acts on an object is called its Center of Gravity. The point where the force of gravity (weight) acts on an object is called its Center of Gravity. For the purpose of this course, center of gravity and the center of mass will be the same thing. For the purpose of this course, center of gravity and the center of mass will be the same thing. However, for very large objects these points may be in slightly different places. However, for very large objects these points may be in slightly different places. Example: the Empire State building has a center of mass that is 1mm higher than its center of gravity. Example: the Empire State building has a center of mass that is 1mm higher than its center of gravity.

4. Center of Mass

5. All projectiles will follow a parabolic motion through the sky. All projectiles will follow a parabolic motion through the sky. However, even if an object explodes in the air, its center of mass will continue to follow this motion. However, even if an object explodes in the air, its center of mass will continue to follow this motion. Example: A firework is launched into the air and explodes at its highest point. Example: A firework is launched into the air and explodes at its highest point.

6. Center of Mass The center of mass of an object can be located many different ways. The center of mass of an object can be located many different ways. Any time you flip an object of any shape or size, it rotates around its center of mass. Any time you flip an object of any shape or size, it rotates around its center of mass. When you balance an object on a point, this point must be below the object’s center of mass. When you balance an object on a point, this point must be below the object’s center of mass.

7. Lesson #66 Topic: Lab: Finding C of G Objectives: (After this class I will be able to) 1. Devise a way to find the exact location of an object’s center of mass. 4/19/07 Warm Up: How would you estimate the location of the center of mass of an object? Trace 5 objects on separate sheets of paper and mark where the center of mass is for each object. Lab task: Trace 5 objects on separate sheets of paper and mark where the center of mass is for each object.

8. Project

9. Lesson #67 Topic: Solving for CM Objectives: (After this class I will be able to) 1. Calculate and solve for the exact location of the center of mass of an object. 4/20/07 Warm Up: Where is the approximate correct location of the center of mass of a car? Assignment: “Solving CG” due at the end of the period tomorrow.

10. Where is the approximate correct location of the center of mass of a car? 1. A 2. B 3. C 4. D 0 of 5

11. Solving for CM / CG The location of the center of mass of a system can be found using the conservation of momentum. The location of the center of mass of a system can be found using the conservation of momentum. Consider a block that explodes into a bunch of pieces. Consider a block that explodes into a bunch of pieces. M = m 1 + m 2 + m 3 + m 4 M = m 1 + m 2 + m 3 + m 4 m1m1 m2m2 m3m3 m4m4 M

12. Solving for CM / CG The momentum of the system before exploding is the same after exploding. The momentum of the system before exploding is the same after exploding. p before =p after p before =p after Mv cm =m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 Mv cm =m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 Remember v=d/t or x/t where x=distance Remember v=d/t or x/t where x=distance M(x cm /t) = m 1 (x 1 /t) + m 2 (x 2 /t) + m 3 (x 3 /t) + m 4 (x 4 /t) M(x cm /t) = m 1 (x 1 /t) + m 2 (x 2 /t) + m 3 (x 3 /t) + m 4 (x 4 /t) Time is the same for the whole system. Time is the same for the whole system.

13. Example Three masses, 2kg, 3kg, and 6kg, are located at positions (3,0), (6,0), and (-4,0) respectively, in meters from the origin. Where is the center of mass of the system? Three masses, 2kg, 3kg, and 6kg, are located at positions (3,0), (6,0), and (-4,0) respectively, in meters from the origin. Where is the center of mass of the system?

14. Three people of roughly equivalent mass m on a lightweight banana boat sit along the x axis at positions x 1 =1m, x 2 =5m, and x 3 =6m. Find the position of the CM. 1. 2m 2. 3m 3. 4m 4. 5m 0 of 5

15. The centers of a 4kg sphere and a 7.5kg sphere are separated by a distance of 1.5m. Where is the center of mass of the system? 1..75m from the 4kg sphere 2..86m from the 4kg sphere 3..98m from the 4kg sphere 4. 1.02m from the 4kg sphere 0 of 5

16. Two skaters with masses of 65kg and 45kg, respectively, stand 8m apart, each holding an end of a rope. The light skater pulls on the rope until the two skaters meet. How far does the light skater travel? 1. 8m 2. 6.5m 3. 4.73m 4. 3.27m 0 of 5

17. 2D Example Find the location of the CM Find the location of the CM m 1 =1kg, m 2 =2kg, m 3 =3kg, m 4 =4kg m 1 =1kg, m 2 =2kg, m 3 =3kg, m 4 =4kg (0,0)(4,0) (0,4) (4,4) x y m1m1 m2m2 m3m3 m4m4

18. Project Try to put your head against the wall and stand up with the chair in the front of the room. Try to put your head against the wall and stand up with the chair in the front of the room. Then try to stand with your heels against the wall and lean over and touch your toes. Then try to stand with your heels against the wall and lean over and touch your toes. Why can’t you do either of these things? Why can’t you do either of these things?

19. Lesson #68 Topic: Stability Objectives: (After this class I will be able to) 1. Define stability. 2. Describe an example of an object that is stable, unstable, and neutral. 4/20/07 Warm Up: What is an example of an object that is stable, unstable, and neutral? Assignment: Concept Development 10 - 1 Book Problems p147 RQ #16, 17, 20, TE 3, 5

20. Stability Stable: An object that is stable is one where any rotation in any direction will raise the object’s center of gravity. Stable: An object that is stable is one where any rotation in any direction will raise the object’s center of gravity.

21. Stability Unstable: An object that is unstable is one where any rotation in any direction will lower the object’s center of gravity. Unstable: An object that is unstable is one where any rotation in any direction will lower the object’s center of gravity.

22. Stability Neutral: An object that is neutral is one where any rotation in any direction will not change the object’s center of gravity. Neutral: An object that is neutral is one where any rotation in any direction will not change the object’s center of gravity.

23. Which of the following objects is stable? 1. A 2. B 3. C 0 of 5

24. Which of the following objects is unstable? 1. A 2. B 3. C 0 of 5

25. Which of the following objects is neutral? 1. A 2. B 3. C 0 of 5