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Chapter 9: Linear Momentum Topics in Chapter 9: • the momentum of a particle, and how the net force acting on a particle causes its momentum to change. • the circumstances under which the total momentum of a system of particles is constant (conserved). • elastic, inelastic, and completely inelastic collisions. • what’s meant by the center of mass of a system, and what determines how the center of mass moves. • how to analyze situations such as rocket propulsion in which the mass of a body changes as it moves. Momentum – a new tool for solving problems. • In many situations, such as a bullet hitting a carrot, we cannot use Newton’s second law to solve problems because we know very little about the complicated forces involved. • In this chapter, we shall introduce momentum and impulse, and the conservation of momentum, to solve such problems. Momentum and Newton’s second law • The momentum of a particle is the product of its mass and its velocity: • Newton’s second law can be written in terms of momentum: An isolated system • Two astronauts push each other as they float freely in the zero-gravity environment of space. • There are no external forces; when this is the case, we have an isolated system. Conservation of momentum • External forces (the normal force and gravity) act on the skaters shown, but their vector sum is zero. • Therefore the total momentum of the skaters is conserved. • Conservation of momentum: If the vector sum of the external forces on a system is zero, the total momentum of the system is constant. Remember that momentum is a vector! • When applying conservation of momentum, remember that momentum is a vector quantity! • Use vector addition to add momenta, as shown at the right. Q9.1 A 3.00kg rifle fires a 0.00500kg bullet at a speed of 300 m/s. Which force is greater in magnitude: the force that the rifle exerts on the bullet or the force that the bullet exerts on the rifle? A. The force that the rifle exerts on the bullet is greater. B. The force that the bullet exerts on the rifle is greater. C. Both forces have the same magnitude. D. The answer depends on how the rifle is held. E. The answer depends on how the rifle is held and on the inner workings of the rifle. A9.1 A 3.00kg rifle fires a 0.00500kg bullet at a speed of 300 m/s. Which force is greater in magnitude: the force that the rifle exerts on the bullet or the force that the bullet exerts on the rifle? A. The force that the rifle exerts on the bullet is greater. B. The force that the bullet exerts on the rifle is greater. C. Both forces have the same magnitude. D. The answer depends on how the rifle is held. E. The answer depends on how the rifle is held and on the inner workings of the rifle. Momentum and Newton’s second law • The momentum of a particle is the product of its mass and its velocity: • Newton’s second law can be written in terms of momentum: Collisions • A collision is a brief, intense interaction between objects. • The collision time is short compared with the timescale of the objects' overall motion. • Internal forces of the collision are so large that we can neglect any external forces acting on the system during the brief collision time. • Therefore linear momentum is essentially conserved during collisions. Elastic and Inelastic Collisions • In an elastic collision kinetic energy is conserved. • Therefore, the internal forces in an elastic collision must be conservative. • In an inelastic collision, the forces are not conservative and mechanical energy is lost. • In a totally inelastic collision, the colliding objects stick together to form a single composite object. • Even if a collision is totally inelastic, that doesn't necessarily mean that all kinetic energy is lost (momentum must still be conserved). Elastic collisions: Before • In an elastic collision, the total kinetic energy of the system is the same after the collision as before. Elastic collisions: During • In an elastic collision, the total kinetic energy of the system is the same after the collision as before. Elastic collisions: After • In an elastic collision, the total kinetic energy of the system is the same after the collision as before. Elastic collisions • Billiard balls deform very little when they collide, and they quickly spring back from any deformation they do undergo. • Hence the force of interaction between the balls is almost perfectly conservative, and the collision is almost perfectly elastic. Elastic collisions in one dimension • Let’s look at a one-dimensional elastic collision between two bodies A and B, in which all the velocities lie along the same line. • We will concentrate on the particular case in which body B is at rest before the collision. • This type of collision conserves both kinetic energy and momentum. Elastic collisions in one dimension • Let’s look at a one-dimensional elastic collision between two bodies A and B, in which all the velocities lie along the same line. • We will concentrate on the particular case in which body B is at rest before the collision. • Conservation of kinetic energy and momentum give the result for the final velocities of A and B: Elastic collisions in one dimension • When B is much more massive than A, then A reverses its velocity direction, and B hardly moves. Elastic collisions in one dimension • When B is much more massive than A, then A reverses its velocity direction, and B hardly moves. Elastic collisions in one dimension • When A and B have similar masses, then A stops after the collision and B moves with the original speed of A. Completely inelastic collisions: Before • In an inelastic collision, the total kinetic energy after the collision is less than before the collision. Completely inelastic collisions: During • In an inelastic collision, the total kinetic energy after the collision is less than before the collision. Completely inelastic collisions: After • In an inelastic collision, the total kinetic energy after the collision is less than before the collision. Inelastic collisions • Cars are designed so that collisions are inelastic—the structure of the car absorbs as much of the energy of the collision as possible. • This absorbed energy cannot be recovered, since it goes into a permanent deformation of the car. Inelastic Collisions Example: Ballistic pendulum. The ballistic pendulum is a device used to measure the speed of a projectile, such as a bullet. The projectile, of mass m, is fired into a large block of mass M, which is suspended like a pendulum. As a result of the collision, the pendulum and projectile together swing up to a maximum height h. Determine the relationship between the initial horizontal speed of the projectile, v, and the maximum height h. Impulse • The impulse of a force is the product of the force and the time interval during which it acts: • On a graph of versus time, the impulse is equal to the area under the curve: Impulse Impulse Impulse and momentum • Impulse–momentum theorem: The change in momentum of a particle during a time interval is equal to the impulse of the net force acting on the particle during that interval: Impulse and momentum • When you land after jumping upward, your momentum changes from a downward value to zero. • It’s best to land with your knees bent so that your legs can flex. • You then take a relatively long time to stop, and the force that the ground exerts on your legs is small. • If you land with your legs extended, you stop in a short time, the force on your legs is larger, and the possibility of injury is greater. Center of mass • We can restate the principle of conservation of momentum in a useful way by using the concept of center of mass. • Suppose we have several particles with masses m1, m2, and so on. • We define the center of mass of the system as the point at the position given by: Center of Mass For two particles, the center of mass lies closer to the one with the most mass: where M is the total mass. Center of mass of symmetrical objects Center of mass of symmetrical objects Motion of the center of mass • The total momentum of a system is equal to the total mass times the velocity of the center of mass. • The center of mass of the wrench at the right moves as though all the mass were concentrated there. Motion of the Center of Mass • The center of mass obeys Newton's second law: Fnet external M acm • The parts of a rotating projectile undergo complex motions, but its center of mass follows a parabolic arc. Center of Mass and Translational Motion The total momentum of a system of particles is equal to the product of the total mass and the velocity of the center of mass. The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration of the center of mass: Therefore, the center of mass of a system of particles (or objects) with total mass M moves like a single particle of mass M acted upon by the same net external force. Center of Mass and Translational Motion • In the absence of any external forces on a system, the center of mass motion remains unchanged; if it's at rest, it remains in the same place—no matter what internal forces may act. • Example: Jumbo, a 4.8-t elephant, walks 19 m toward one end of the car, but the CM of the 15-t rail car plus elephant doesn't move. This allows us to find the car's final position (we set the initial position of the car's CM at the origin): Question1: Conservation of Momentum Falling on or off a sled. An empty sled is sliding on frictionless ice when Susan drops vertically from a tree above onto the sled. When she lands, does the sled: A) speed up B) slow down, or C) keep the same speed? Question1: Conservation of Momentum Falling on or off a sled. An empty sled is sliding on frictionless ice when Susan drops vertically from a tree above onto the sled. When she lands, does the sled: A) speed up B) slow down, or C) keep the same speed? Question 2: When a cannonball is fired from a cannon, why does the cannon recoil backward? a) The total energy of the cannonball and cannon is conserved. b) The energy of the cannon is greater than the energy of the cannonball. c) The momentum of the cannon is conserved. d) The momentum of the cannon is greater than the momentum of the cannonball. e) The total momentum of the cannonball and cannon is conserved. Question 2: When a cannonball is fired from a cannon, why does the cannon recoil backward? a) The total energy of the cannonball and cannon is conserved. b) The energy of the cannon is greater than the energy of the cannonball. c) The momentum of the cannon is conserved. d) The momentum of the cannon is greater than the momentum of the cannonball. e) The total momentum of the cannonball and cannon is conserved. Question 3 Two objects with different masses collide with and stick to each other. Compared to before the collision, the system of two objects after the collision has A B A. the same amount of total momentum and the same total kinetic energy. B. the same amount of total momentum but less total kinetic energy. C. less total momentum but the same amount of total kinetic energy. D. less total momentum and less total kinetic energy. E. Not enough information is given to decide. Question 3 Two objects with different masses collide with and stick to each other. Compared to before the collision, the system of two objects after the collision has A B A. the same amount of total momentum and the same total kinetic energy. B. the same amount of total momentum but less total kinetic energy. C. less total momentum but the same amount of total kinetic energy. D. less total momentum and less total kinetic energy. E. Not enough information is given to decide. Summary of Chapter 9 • Momentum of an object: • Newton’s second law: • Total momentum of an isolated system of objects is conserved. • During a collision, the colliding objects can be considered to be an isolated system even if external forces exist, as long as they are not too large. • Momentum will therefore be conserved during collisions. Summary of Chapter 9, cont. • Impulse: • In an elastic collision, total kinetic energy is also conserved. • In an inelastic collision, some kinetic energy is lost. • In a completely inelastic collision, the two objects stick together after the collision. • The center of mass of a system is the point at which external forces can be considered to act.