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Chapter 9 Systems of Particles The Center of Mass

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Chapter 9 Systems of Particles The Center of Mass

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    1. 1 Chapter 9 – Systems of Particles The Center of Mass The center of mass of a body or a system of bodies is the point that moves as though all of the mass were concentrated there and all external forces were applied there.

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    3. 3 The center of mass of an object needs not lie within the object. There is no dough at the center of mass of a doughnut, and no iron at the center of mass of a horseshoe.

    4. 4 Sample Problem 9-1 Three particles of masses m1 = 1.2 kg, m2 = 2.5 kg, and m3 = 3.4 kg form an equilateral triangle of edge length a = 140 cm. Where is the center of mass of this three-particle system?

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    7. 7 Newton’s Second Law for a System of Particles is the net force of all external forces that act on the system. Forces on one part of the system from another (internal forces) are not included. M is the total mass of the system. We assume that no mass enters or leaves the system as it moves, so that M remains constant. The system is said to be closed. is the acceleration of the center of mass of the system. The equation gives no information about the acceleration of any other point of the system.

    8. 8 This means that the center of mass of the fragments follows the same parabolic trajectory that the rocket would have followed had it not exploded.

    9. 9 Sample Problem 9-3 The three particles in the figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are F1 = 6.0 N, F2 = 12 N, and F3 = 14 N. What is the acceleration of the center of mass of the system, and in what direction does it move?

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    12. 12 Linear Momentum Linear momentum is a vector and the SI unit is kg-m / s. For constant mass,

    13. 13 Linear Momentum for a System of Particles The linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass. For constant mass,

    14. 14 Sample Problem 9-4 Figure 9-8a shows a 2.0 kg toy race car before and after taking a turn on a track. Its speed is 0.50 m/s before the turn and 0.40 m/s after the turn. What is the change D in the linear momentum of the car due to the turn?

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    16. 16 Conservation of Linear Momentum If no net external force acts on a system of particles, the total linear momentum of the system cannot change. If the component of the net external force on a closed system is zero along an axis, then the component of the linear momentum of the system along that axis cannot change.

    17. 17 Sample Problem 9-5 A ballot box with mass m = 6.0 kg slides with speed v = 4.0 m/s across a frictionless floor in the positive direction of an x axis. It suddenly explodes into two pieces. One piece, with mass m1 = 2.0 kg, moves in the positive direction of the x axis with speed v1 = 8.0 m/s. What is the velocity of the second piece, with mass m2 ?

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    19. 19 Sample Problem 9-7 A firecracker placed inside a coconut of mass M, initially at rest on a frictionless floor, blows the coconut into three pieces that slide across the floor. An overhead view is shown in the figure. Piece C, with mass 0.30M, has final speed vfC = 5.0 m/s. (a)  What is the speed of piece B, with mass 0.20M?

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    21. 21 (b)  What is the speed of piece A?

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