Chapter 9 Linear Momentum and Collisions

Transcription

1 Chapter 9 Linear Momentum and Collisions

2 The Center of Mass The center of mass of a system of particles is the point that moves as though (1) all of the system s mass were concentrated there and (2) all external forces were applied there. 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.

3 9-7 Center of Mass The center of mass of a system is the point where the system can be balanced in a uniform gravitational field.

4 The Center of Mass: A System of Particles Consider a situation in which n particles are strung out along the x axis. Let the mass of the particles are m 1, m 2,.m n, and let them be located at x 1, x 2, x n respectively. Then if the total mass is M = m 1 + m m n, then the location of the center of mass, x com, is

5 For two objects: 9-7 Center of Mass The center of mass is closer to the more massive object.

6 The Center of Mass: A System of Particles In 3-D, the locations of the center of mass are given by: The position of the center of mass can be expressed as:

7 9-7 Center of Mass The center of mass need not be within the object:

8 Sample problem, COM of 3 particles The total mass M of the system is 7.1 kg. The coordinates of the center of mass are therefore: We are given the following data: Note that the z com = 0.

9 Newton s 2 nd Law for a System of Particles The vector equation that governs the motion of the center of mass of such a system of particles is: Note that: 1. F net is the net force of all external forces that act on the system. Forces on one part of the system from another part of the system (internal forces) are not included 2. M is the total mass of the system. M remains constant, and the system is said to be closed. 3. a com is the acceleration of the center of mass of the system.

10 Newton s 2 nd Law for a System of Particles: Proof of final result For a system of n particles, where M is the total mass, and r i are the position vectors of the masses m i. Differentiating, where the v vectors are velocity vectors. This leads to Finally, What remains on the right hand side is the vector sum of all the external forces that act on the system, while the internal forces cancel out by Newton s 3 rd Law.

11 9-7 Center of Mass Motion of the center of mass:

12 Sample problem: motion of the com of 3 particles Calculations: Applying Newton s second law to the center of mass,

13 9-1 Linear Momentum Momentum is a vector; its direction is the same as the direction of the velocity.

14 9-1 Linear Momentum Change in momentum: (a) mv (b) 2mv

15 Linear momentum DEFINITION: in which m is the mass of the particle and v is its velocity. The time rate of change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force. Manipulating this equation: (Newton s 2 nd Law)

16 Linear Momentum of 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.

17 9-2 Momentum and Newton s Second Law Newton s second law, as we wrote it before: is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:

18 Collision and Impulse In this case, the collision is brief, and the ball experiences a force that is great enough to slow, stop, or even reverse its motion. The figure depicts the collision at one instant. The ball experiences a force F(t) that varies during the collision and changes the linear momentum of the ball.

19 9-3 Impulse Therefore, the same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.

20 Collision and Impulse Instead of the ball, one can focus on the bat. At any instant, Newton s third law says that the force on the bat has the same magnitude but the opposite direction as the force on the ball. That means that the impulse on the bat has the same magnitude but the opposite direction as the impulse on the ball.

21 9-3 Impulse Impulse is a vector, in the same direction as the average force.

22 Sample problem: 2-D impulse

23 Sample problem: 2-D impulse, cont.

24 Conservation of Linear Momentum If no net external force acts on a system of particles, the total linear momentum, P, 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.

25 9-4 Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change:

26 9-4 Conservation of Linear Momentum Internal Versus External Forces: Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it.

27 9-4 Conservation of Linear Momentum Furthermore, internal forces cannot change the momentum of a system. However, the momenta of components of the system may change.

28 Sample problem: conservation of momentum The collision within the bullet block system is so brief. Therefore: (1)During the collision, the gravitational force on the block and the force on the block from the cords are still balanced. Thus, during the collision, the net external impulse on the bullet block system is zero. Therefore, the system is isolated and its total linear momentum is conserved. (2) The collision is one-dimensional in the sense that the direction of the bullet and block just after the collision is in the bullet s original direction of motion. As the bullet and block now swing up together, the mechanical energy of the bullet block Earth system is conserved: Combining steps:

29 Momentum and Kinetic Energy in Collisions In a closed and isolated system, if there are two colliding bodies, and the total kinetic energy is unchanged by the collision, then the kinetic energy of the system is conserved (it is the same before and after the collision). Such a collision is called an elastic collision. If during the collision, some energy is always transferred from kinetic energy to other forms of energy, such as thermal energy or energy of sound, then the kinetic energy of the system is not conserved. Such a collision is called an inelastic collision.

30 Inelastic collisions in 1-D An inelastic collisions occurs when two objects collide and do not bounce away from each other. Momentum is conserved, because the total momentum of both objects before and after the collision is the same. However, kinetic energy is not conserved. Some of the kinetic energy is converted into sound, heat, and deformation of the objects. A high speed car collision is an inelastic collision.

31 Inelastic Collisions A completely inelastic collision:

32 9-5 Inelastic Collisions For collisions in two dimensions, conservation of momentum is applied separately along each axis:

33 Inelastic collisions in 1-D

34 Elastic collisions in 1-D An elastic collision occurs when the two objects "bounce" apart when they collide. Two rubber balls are a good example. In an elastic collision, both momentum and kinetic energy are conserved. Almost no energy is lost to sound, heat, or deformation. The first rubber ball deforms, but then quickly bounces back to its former shape, and transfers almost all the kinetic energy to the second ball.

35 Elastic collisions in 1-D In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.

36 Elastic collisions in 1-D A car's bumper works by using this principle to prevent damage. In a low speed collision, the kinetic energy is small enough that the bumper can deform and then bounce back, transferring all the energy directly back into motion. Almost no energy is converted into heat, noise, or damage to the body the body of the car, as it would in an inelastic collision. However, car bumpers are often made to collapse if the speed high enough, and not use the benefits of an elastic collision. The rational is that if you are going to collide with something at a high speed, it is better to allow the kinetic energy to crumple the bumper in an elastic collision than let the bumper you around as your car bounces in an elastic collision.

37 Elastic collisions in 1-D: Stationary Target

38 Elastic collisions in 1-D: Moving Target

39 Sample problem: two pendulums