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Announcements

Announcements. Quiz 4 on Friday Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy ). Lecture 10: Center of Mass. Today’s Concept: Center of mass Finding it Using it. Center of Mass. Cube Example. Cube Example: Using Volume Mass Density. ρ = mass density (mass/volume).

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Announcements

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  1. Announcements • Quiz 4 on Friday • Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy)

  2. Lecture 10: Center of Mass Today’s Concept: Center of mass Finding it Using it

  3. Center of Mass

  4. Cube Example

  5. Cube Example: Using Volume Mass Density ρ= mass density (mass/volume)

  6. Cube Example: CM Calculation

  7. Exercise Show mathematically that the center of mass of a uniform stick of length Land mass Mis at x = L/2. dm x = L x = 0

  8. Exercise Show mathematically that the center of mass of a uniform stick of length Land mass Mis at x = L/2. dm x = L x = 0

  9. Center of Mass for System of Objects

  10. ACT: Disk vs. 2 half-disks The disk shown in Case1 clearly has its CMat the center. Suppose the disk is cut in half and the pieces arranged as shown in Case2 In which case is the center of mass highest? A) Case 1 B) Case 2 C) same x xCM Case1 Case2

  11. ACT: Disk vs. 2 half-disks The disk shown in Case1 clearly has its CMat the center. Suppose the disk is cut in half and the pieces arranged as shown in Case2 In which case is the center of mass highest? A) Case 1 B) Case 2 C) same x x x xCM x Case1 Case2

  12. Kinematic Quantities If the net external force is zero, the CMwon’t accelerate

  13. ACT: System of particles Three tiny equal-mass magnets are placed on a horizontal frictionless surface at the corners of an equilateral triangle. When the magnets are released, they attract and quickly slide to a single point. What are the coordinates of that point? A) 0 0 B) 0 H/2 C) 0 H/3 D) H/4 H/4 E) H/4 0 y x y H x

  14. C) 0 H/3 y X Y H (symmetry) x

  15. Homework Problem = 0.975 m = -1 m

  16. Homework Problem

  17. CM moves toward m5 ABCD m5 cm

  18. CM moves toward m5 ABC m5 cm

  19. CheckPoint: System of Two Masses Two objects, one having twice the mass of the other, are initially at rest. Two forces, one twice as big as the other, act on the objects in opposite directions as shown. 2F F M 2M Which of the following statements about the acceleration of the center of mass of the system is true? A) ACM= F/M to the right B) ACM= F/(3M) to the right C) ACM= 0 D) ACM= F/(3M) to the left E) ACM= F/M to the left About 42% answered correctly.

  20. CheckPointRevote: System of Two Masses 2F F M 2M Which of the following statements about the acceleration of the center of mass of the system is true: C) ACM= 0 D) ACM= F/(3M) to the left E) ACM= F/M to the left C) The actual masses are accelerating away from each other at the same rate so the center of mass does not change. D) CM acceleration is defined as the sum of the total forces divided by the total mass. Total mass is 3m. The forces are not equal, therefore the greater force will determine the direction of acceleration. E) The acceleration of the center of mass is equal to the external force divided by the total mass.

  21. ACT: System of 2 Masses Two objects, one having twice the mass of the other, are initially at rest. A force Fpushes the mass Mto the right. A force pushes the mass 2M upward. What is the acceleration of the center of mass? A) B)C)D) M 2M F

  22. CheckPoint: Two Men Pulling Two guys who weigh the same are holding onto a masslesspole while standing on horizontal frictionless ice. If the guy on the left starts to pull on the pole, where do they meet? -3 m 0 m 3 m A) -3 m B) 0 m C) 3 m Where is the center of mass of the system initially? Does the center of mass accelerate?

  23. ACT: 2 guys pulling, II M 2M A large skinny guy with mass 2Mand a smaller guy with mass Mare holding onto a masslesspole while standing on frictionless ice, as shown below. If the little guy pulls himself toward the big guy, where would they meet? A) -3 m B) -1 m C) 0 m D) 1 m E) 3 m -3 m 0 m 3 m Where is the center of mass of the system initially? Does the center of mass accelerate?

  24. M 2M ACT: 2 guys pulling, II A large skinny guy with mass 2Mand a smaller guy with mass Mare holding onto a masslesspole while standing on frictionless ice, as shown below. If the little guy pulls himself toward the big guy, where would they meet? A) -3 m B) -1 m C) 0 m D) 1 m E) 3 m -3 m 0 m 3 m m m = -1 m

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