2. Centre of Mass Created by J Harris Jharris.school@gmail.com

3. Centre of Mass

4. Find the mass of a random 2D shape: Cut of a random 2D shape out of paper / card Put a pin through your card into the wall, at an edge of your card. Hang a plumb bob from the pin. Trace the line the string from the plumb bob makes Change the position of the pin, and trace the new line the string makes Can you find the centre of mass?

5. Find the mass of a random 2D shape: Start with any arbitrary shape

6. Find the mass of a random 2D shape: An arbitrary shape The line of a plumb bob from the edge of the shape The line of a plumb bob from a different edge gives the location of the centre of mass

7. 24-May-15 Physics 1 (Garcia) SJSU Locating Center of Gravity Balance an object to find center of gravity Center of Gravity

8. Centre of Mass: The centre of mass (CoM) is an imaginary point that can lie either inside (e.g. a shot put) or outside a body (hula hoop). A system’s mass responds to external forces and torques, as if its entire mass were concentrated about that point.

9. Where is the COM? The position of centre of mass depends on the shape of body This is how the high jumper can have his CoM pass under the bar but he could still clear the bar.

10. CoM depends on shape

11. Where is the COM?

12. 24-May-15 Physics 1 (Garcia) SJSU Demo: Balanced Bird Where is the bird’s center of gravity?

13. Try this one at home!

14. 24-May-15 Physics 1 (Garcia) SJSU Check Yourself Three trucks are parked on a slope. Which truck(s) tip over? CG BASE

15. Pictures of motion of COM

16. Centre of Mass & Balance What 3 mechanical principles enable a body to become more stable? The lower the C.O.M. the more stable the body The wider the base of support the more stable the body The closer the line of gravity to the centre of the base of support, the more stable the body

17. 24-May-15 Physics 1 (Garcia) SJSU Demo: Balance the Can If a small amount of water is added to an empty soda can then the can may be balanced on its bottom edge. x CG PEPSI

18. 24-May-15 Physics 1 (Garcia) SJSU Stability Object is stable if CG is above the base. CG Weight BASE Axis BASE STABLE UNSTABLE

19. What is the difference between COG & COM? The term Centre of Mass is often confused with the Centre of Gravity. The two terms are so similar that they can be used interchangeably. The COG of an object coincides with its COM if the object is in a completely uniform gravitational field. If it is not in a uniform gravitational field, then the COG and the COM will be at too different locations.

20. Finding CoM in 2D

21. m1m1 m2m2 m1m1 m2m2 m1m1 m2m2 m1m1 m2m2 m1m1 m2m2 m1gm1gm2gm2g d1d1 d2d2 M = m 1 + m 2 The Centre of Mass CM M m 1 g x d 1 = m 2 g x d 2

22. Centre of Mass (1D) 0 x1x1 x2x2 x cm m1m1 CoM m2m2 M = m 1 + m 2 M x cm = m 1 x 1 + m 2 x 2

23. See pic on pg 35 of rutter

24. Why study Center of Mass? (a) A ball tossed into the air follows a parabolic path. (b) The center of mass (black dot) of a baseball bat flipped into the air follows a parabolic path, but all other points of the bat follow more complicated curved paths.

25. The motion of the Centre of Mass is a simple parabola. (just like a point particle) The motion of the entire object is complicated. This motion resolves to motion of the CM The Centre of Mass motion of points around the CM

26. Types of Collisions Momentum is conserved in any collision Inelastic collisions: rubber ball and hard ball –Kinetic energy is not conserved –Perfectly inelastic collisions occur when the objects stick together Elastic collisions: billiard ball –both momentum and kinetic energy are conserved Actual collisions –Most collisions fall between elastic and perfectly inelastic collisions

27. Conservation of Momentum In an isolated and closed system, the total momentum of the system remains constant in time. –Isolated system: no external forces –Closed system: no mass enters or leaves –The linear momentum of each colliding body may change. –The total momentum P of the system cannot change.

28. Motion of the Centre of Mass The complex motion of many separate masses can often be made clear by only considering the single motion of the CoM. Even though the separate masses may collide and interact, the velocity of the CoM v cm,does not change, assuming no external forces are present.

29. Motion of the Centre of Mass The velocity of the centre of mass can be found by adding up all the individual momentums. Then to find the velocity use p=mv. So divide by the total mass will give the ‘average / overall velocity’ called the velocity of the centre of mass.

30. http://www.youtube.com/watch?v=gPDjaMsG_rg

31. Centre of Mass (3D) For a collection of masses in 3D cm r1r1 r3r3 r2r2

32. y x0 2 21 1 3 kg 8 kg 4 kg X cm = 16/15 = 1.07 m y cm = 20/15 = 1.33 m 15 kg r CM 1.07 1.33