2. Where is the center or mass of a wrench?

3.  Center-of-Mass A mechanical system moves as if all of its mass were concentrated at this point A very special point used to describe the overall motion of a system

4. Projectiles  Motion can be described in two ways:  Rotation about a special point  Parabolic curve traced by that same point

5. Where is the center of mass of a ball?

6. Where is the center or mass of a barbell?

7. Binary Stars

8. CoM  The weighted average position of the system’s mass Capital M is the sum of the masses. In 3-D, you have to calculate it 3 times ; once along the x-axis, then y, then Z. This gives you the (x,y,z coordinates of the center of mass)

9. Complex, asymetric bodies of uniform density can be simplified by breaking them into easily identified, symmetric pieces and considering all of the mass of that piece to be at it’s geometric center. L/2 L Answer= (0.5ML, 0.3ML)

10. Complex, asymetric bodies of uniform density can be simplified by breaking them into easily identified, symmetric pieces and considering all of the mass of that piece to be at it’s geometric center. L/2 L CMx = M·0+3M·(L/2) + M·(L) 5M CMy = M·0+3M·(L/2) + M·(0) 5M Answer= (0.5ML, 0.3ML)

11. 1.Chop it into manageable pieces. 2.Locate their CMs by symmetry 3.Fix an origin 4.Find CM of whole system by finding the x and y distance to each little CM and plugging into the CM formula. We’ve now created a 5-body problem that is easier to handle than the original whole shape by taking advantage of symmetries.

12. 1.Chop it into manageable pieces. 2.Locate their CMs by symmetry 3.Fix an origin 4.Find CM of whole system by finding the x and y distance to each little CM and plugging into the CM formula. We’ve now created a 5-body problem that is easier to handle than the original whole shape by taking advantage of symmetries.

13. Answer: x = - 0.25 m, y = 0M\Therefor the missing piece of Q is Q /9 since Area = (2/6) 2. (-1.5,1.5) (-1.5,-1.5) (1.5,2) (-1.5, -2) (0.5,0) ¼ Q(-1.5) + ¼Q (-1.5) + 1/6 Q(1.5) + 1/6Q (1.5) + 1/18Q(0.5) __________________________________________ Q 8/9 Call the total mass before the cut out “Q”

14. CoM  The CoM can also be defined by its position vector, which starts at the origin and goes to the CM point. R CM

15. Example Consider the following masses and their coordinates which make up a "discrete mass" or “rigid body”. What are the coordinates for the center of mass of this system?

16. -0.94 0.19 4.4

17. Locating the CoM of an object without calculation Throw it, watch for the point it spins about. Hang it from two different points and find the intersection The CoM of a symmetric object lies on an axis of symmetry and on any plane of symmetry Balance it on your finger

18. The CoM is a.k.a. the Center of Gravity  Each element of mass making up an object is acted upon by gravity.  The CoG is another special point where all gravitational force acts (Mg).  This is the same location as the CoM if g is constant over the object.  If an object is pivoted at is CoG, it balances in any orientation

19. Where is the Center of Gravity (CG)?  CG is at the midpoint for uniform objects.  CG is the balance point.  CG will be directly below a single point of suspension.  CG may exist where these is no actual material.

20. Old Physics formulas applied to the CoM momentum Take the time derivative of the position of the center of mass The total linear momentum of the system equals the total mass multiplied by the velocity of the CoM.

21. Fireworks”: What does the CoM do? See p. 193

22. Take another time derivative Acceleration of the Center of Mass The CoM moves like an imaginary particle of mass M under the influence of the resultant external force on the system.

23. Where is the CG? In the head In the air in the center

24. Where is the CG? In the center air closer to the diamond In the hole!

25. Toppling  If the CG of an object is above the area of support, the object will remain upright.  If the CG extends outside the area of support, the object will topple.

26. Stability  An object with a low CG is usually more stable than an object with a high CG.

27. Equilibrium  Unstable equilibrium  An object balanced so that any displacement lowers its center of gravity.  Stable equilibrium  Any displacement raises its CG  Neutral equilibrium  CG is neither raised or lowered with displacement

28. CoM  For odd-shaped, extended objects with continuous mass distribution? Why is this only approximate? We must let the number of mass elements approach infinity so this is not an approximation. Replace the sum with an integral and the element Δm with differential element dm: