1. If an asymmetrical object, such as an axe, is tossed, its motion seems rather complicated, but one special point moves in a simple __________ path… This point is called the _______ ___ _____. parabolic Center of Mass

2. Center of Mass vs. Center of Gravity Are the two terms, Center of Mass (CM) and Center of Gravity (CG) synonymous? If gravity may be considered _________, then _____! For a tall object, such as a skyscraper, the CG would be _________________ the CM, as the pull of earth’s gravity on the top of the building is slightly ____ than on the bottom. uniformYES just slightly below less

3. Definition of the Center of Mass (CM) The center of mass of a body or of a system of particles is the point that moves as though _________________ were concentrated there, and any external _______ were applied there. It can be experimentally determined by finding the place where the body/system __________. For a system of particles, the center of mass is defined as the _________ of their positions, weighted by their ________. all of the mass forces balances average masses

4. Calculating the Center of Mass (CM) Likewise for Z cm if the body or system of particles has 3 dimensions.

5. Example 1: Find the CM of the system shown at right The answer is quite logical, as the 2 kg mass is ______ of the total mass. 2/3rds

6. Lotso’ Center of Gravity Examples: 1. Demo: The Balancing Bird The CG of the bird is located ___________. If the CG is __________, it will not fall !! Why does the bird balance? at its beak supported

7. Demo: Touch your toes while standing against a wall… Why can’t it be done? A _______ is created when the CG extends beyond the _______ (_____) causing the person to rotate. How could it be done? Extend the support area! torque support feet

8. Why does a pregnant woman often experience back pain? As she gains weight, her CG _______________, and eventually would extend beyond her support (shoes) Is there a solution? ______(_____ _______) to extend the support or a ________ to shift the CG back. Drawings from Paul Hewitt’s Conceptual Physics shifts outward skis clown shoes?? backpack

9. Demo: Stand 2 footlengths away from a wall, bend over, touch your head to the wall, pick up a chair, and try to stand up! This is typically easier for ________ than ______ because females usually have _______ _____ and a _______ ____. females males smaller feet lower CG

10. Example 2: Find the CM of a thin piece of metal bent into a right angle that has a total mass, M. Let m 1 = horizontal piece with mass _____. Let m 2 = vertical piece with mass _____. ½m Or use symmetry…

11. Even though the CM is outside the body, if tossed, the CM will follow a parabolic path.

12. Example 3: Find the CM of a uniform triangular shape … using symmetry instead of calculation. When a median is drawn (line segment from a vertex to the opposite ___________), the mass on one side of the line is __________ the mass on the other side. So, the CM must be located at the _____________ of the three medians. To find the medians experimentally, hang the triangle from each vertex… The CM must lie directly _______ the vertex, so the median will be _________. midpoint equal to intersection belowvertical

13. And now an easier way… using the CM…. Treat the portion of the rope hanging over the edge as if it’s all concentrated at its ___. Calculate the work required to lift the CM to the top of the table, a distance of CM Where F = weight of ____ of rope

14. Example 5: A Practical Application.. Before a tire is put on a car, a tire balancing machine is used to ensure that the tire’s ____ coincides with its ________________. If needed, small weights are attached to the rim of the tire to shift the CM to the geometrical center. Typical tire weight CM geometrical center

15. Where M = Total Mass Dividing by the change in time (∆t) we get Rearranging: …. A result stating that the linear __________ of a system of particles is equal to the product of the total mass and the velocity of the center of mass. momentum

16. From Newton’s 2 nd Law, … Since any internal forces on the system would cancel out (Newton’s 3 rd Law), the vector sum of all the forces on the right side of the equation is equivalent to the Substituting yields… where M = total mass

17. 1. When a firework shell is projected, it takes a parabolic path (neglecting air drag). The forces that cause the firework to explode are ________ to the system. So… the on the system is just the _____________________. Using a cm = __, so the CM of the system will continue to follow _______________________. (until the first fragments hit the ground!) internal gravitational force g the original parabolic path

18. 2. A ballet dancer, leaping in what is termed a “grand jete”, seems to float across the stage. Why? As she leaps, she raises her arms and stretches out her legs. This shifts her CM ___, which __________ the height that the head would attain in a normal jump … making her look like she is “floating” horizontally. upreduces

19. 3. A closer look at orbital motion…. Astronomers have discovered ________ _____ systems that orbit their ____. The position of the CM does not change since there are NO external forces on the system. Similarly, the earth and the moon orbit their ____, which is actually _________________ because the earth is so much more massive than the moon! binary star CM inside the earth

20. Ignoring the other planets, the ___________ ___________________ is the only external force acting on the earth/moon system. This force makes the ____ of the earth/moon take a smooth elliptical (almost circular!) orbit around the ____ of the _______________________. The sun also slightly “wobbles” about this CM. When astronomers observe a star that “wobbles”, that’s evidence that it has __________ orbiting it! (greatly exaggerated) gravitational pull of the Sun CM Sun/Earth/Moon system planets