Outline n n n 9 1 What Is

Outline n n n 9 -1 What Is Physics? 9 -2 The Center of Mass 9 -3 Newton’s Second Law for a System of Particles 9 -4 Linear Momentum 9 -5 The Linear Momentum of a System of Particles 9 -6 Collision and Impulse 9 -7 Conservation of Linear Momentum 9 -8 Momentum and Kinetic Energy in Collisions 9 -9 Inelastic Collisions in One Dimension 9 -10 Elastic Collisions in One Dimension 9 -11 Collisions in Two Dimensions 9 -12 Systems with Varying Mass: A Rocket 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 2

9 -1 What Is Physics? n We will derive the equation of motion for the center of mass, and discuss the principle of conservation of linear momentum. n Finally, we will use the conservation of linear momentum to study collisions in one and two dimensions and derive the equation of motion for rockets. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 3

9 -2 The Center of Mass n 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. (2) all external forces were applied there. n Define the center of mass (com) of particles in order to predict the possible motion of the system. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 4

9 -2 The Center of Mass n Systems of Particles: Define the position of the center of mass (com) of this twoparticle system to be 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 5

9 -2 The Center of Mass n The coordinate system has been shifted leftward. The position of the center of mass is now defined as M: total mass of the system 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 6

9 -2 The Center of Mass n For N particles the (vector) position of the CM is M: total mass of the system 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 7

9 -2 The Center of Mass n In three dimensions, the center of mass must be identified by three coordinates. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 8

9 -2 The Center of Mass n Center 2021/2/20 of Mass of Continuous Bodies: 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 9

9 -2 The Center of Mass n Consider the thin rod of mass M and length L. The mass of the element of volume d. V is : kg/m 3 ; n For : linear mass density a disk or cylinder, : areal mass density ( kg/m 2 ) 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 10

Example 9 -1 n Three particles of masses m 1 = 1. 2 kg, m 2 = 2. 5 kg, and m 3 = 3. 4 kg form an equilateral triangle of edge length a = 140 cm. Where is the center of mass of this system? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 11

Example 9 -2 n Figure shows a uniform metal plate P of radius 2 R from which a disk of radius R has been stamped out (removed) in an assembly line. Using the xy coordinate system shown, locate the center of mass com. P of the plate. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 14

Example 9 -3 n. A thin rod of length 3 L is bent at right angles at a distance L from one end. Locate CM with respect to the corner. Take L = 1. 2 m. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 17

Example 9 -3 n Solution: The CM of each arm is at its midpoint. If we take m 1 = m, then m 2 = 2 m. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 18

9 -3 Newton’s Second Law for a System of Particles n Newton’s second law for the motion of the center of mass of a system of particles: ( System of particles ) (1) is the net force of all external forces that act on the system. (2) M is total mass of the system. (3) is the acceleration of the center of mass. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 19

9 -3 Newton’s Second Law for a System of Particles n Proof z of equation: F 1 m 2 F 2 x 2021/2/20 m 1 F 3 m 3 O 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 y 20

9 -3 Newton’s Second Law for a System of Particles 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 21

9 -3 Newton’s Second Law for a System of Particles n In terms of components, we have n. A fireworks rocket explodes in flight. The center of mass of the fragments would continue to follow the original parabolic path. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 22

Example 9 -4 n The three particles are initially at rest. Each experiences an external force due to bodies outside three-particle system. The directions are indicated, and the magnitudes are F 1 = 6. 0 N, F 2 = 12 N, and F 3 = 14 N. What is the acceleration of the center of mass of the system, and in 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 23

Example 9 -4 what direction does it move? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 24

Example 9 -5 n. A man of mass m 1 = 60 kg is at the rear of a stationary boat of mass m 2 = 40 kg and length 3 m, which can move freely on the water; see Fig. (a). The front of the boat is 2 m from the dock. What happens when the man walks to the front? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 26

Example 9 -5 n Solution: In terms of the initial position, Since there is no external force, x. CM is fixed. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 27

9 -4 Linear Momentum n Linear Momentum: m: mass of a particle V: velocity of a particle n The SI unit is. n 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. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 28

9 -4 Linear Momentum n In equation form this becomes n Proof 2021/2/20 of equation: 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 29

9 -5 The Linear Momentum of a System of Particles n The linear momentum of a system of particles can by changed only by a net force. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 30

9 -6 Collision and Impulse n The ball experiences a force that varies during the collision and changes the linear momentum of the ball. n In time interval dt, the change in the ball’s momentum 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 31

9 -6 Collision and Impulse n In many situations we do know the average force and the duration of the collision. Thus 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 33

9 -6 Collision and Impulse n Series of Collisions : If the particle stop after the collision, then If the particle bounce backward, then 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 34

Example 9 -6 n When a male bighorn sheep runs head-first into an other male, the rate at which its speed drops to zero is dramatic. Figure gives a typical graph of the acceleration a versus time t for such a collision, with the acceleration taken as negative to correspond to an initially positive velocity. The peak acceleration has magnitude 34 m/s 2 and the duration of the collision is 0. 27 s. Assume that the sheep’s mass is 90. 0 kg. What are 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 35

Example 9 -6 the magnitudes of the impulse and average force due to the collision? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 36

Example 9 -7 (a) n Figure is an overhead view of the path taken by a racecar driver as his car collides with the racetrack wall. Just before the collision, he is traveling at speed vi = 70 m/s along a straight line at 30° from the wall. Just after the collision, he is traveling at speed vf = 50 m/s along a straight line at 10° from the wall. His mass m is 80 kg. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 37

Example 9 -7 (a) What is the impulse the collision? n Solution: 2021/2/20 on the driver due to 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 38

Example 9 -7 (b) n The collision lasts for 14 ms. What is the magnitude of the average force on the driver during the collision? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 40

9 -7 Conservation of Linear Momentum n Law of Conservation of linear momentum: If the net external force on a system is zero, the total linear momentum is constant. (closed, isolated system) 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 41

Example 9 -8 n One-dimensional explosion: 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. The box explodes into two pieces. One piece, with mass m 1 = 2. 0 kg, moves in the positive direction of the x axis at v 1 = 8. 0 m/s. What is the velocity of the second piece, with mass m 2? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 42

Example 9 -8 n Solution: The initial momentum The final momenta of the two pieces are 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 43

Example 9 -9 n One-dimensional explosion: Fig. (a) shows a space hauler and cargo module, of total mass M, traveling along an x axis in deep space. They have an initial velocity of magnitude 2100 km/h relative to the Sun. With a small explosion, the hauler ejects the cargo module, of mass 0. 20 M (Fig. (b)). The hauler then travels 500 km/h faster than the module along the x axis; that is, the relative 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 44

Example 9 -9 speed between the hauler and the module is 500 km/h. What then is the velocity of the hauler relative to the Sun? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 45

Example 9 -10 (a) n Two-dimensional explosion: 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 Fig. (a). Piece C, with mass 0. 30 M, 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 47

Example 9 -10 (a) has final speed vf. C = 5. 0 m/s. What is the speed of piece B, with mass 0. 20 M? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 48

Example 9 -10 (b) n What 2021/2/20 is the speed of piece A? 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 49

9 -8 Momentum and Kinetic Energy in Collisions n Type of collision: (1) A perfectly elastic collision is defined as one in which the total kinetic energy of the particles is also conserved: (2) In an inelastic collision, the total kinetic energy of the particles changes. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 50

9 -9 Inelastic Collisions in One Dimension n One-dimensional inelastic collision: Two bodies just before and just after they have a one-dimensional collision. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 51

9 -9 Inelastic Collisions in One Dimension n One-Dimensional Completely Inelastic Collision: Two bodies before and after they have a completely inelastic collision (meaning they stick together). 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 52

9 -9 Inelastic Collisions in One Dimension n Velocity 2021/2/20 of the Center of Mass: 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 53

Example 9 -11 n The ballistic pendulum was used to measure the speeds of bullets before electronic timing devices were developed. The version shown in figure consists of a large block of wood of mass M = 5. 4 kg, hanging from two long cords. A bullet of 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 54

Example 9 -11 mass m = 9. 5 g is fired into the block, coming quickly to rest. The block + bullet then swing upward, their center of mass rising a vertical distance h = 6. 3 cm before the pendulum comes momentarily to rest at the end of its arc. What is the speed of the bullet just prior to the collision? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 55

Example 9 -12 (a) n. A 200 -kg Cadillac limousine moving east at 10 m/s collides with a 1000 -kg Honda Prelude moving west at 26 m/s. The collision is completely inelastic and takes place on an icy (frictionless) patch of road. Find their common velocity after the collision. m 1: Calillac m 2: Honda 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 57

Example 9 -12 (a) n Solution: We assume the unknown common velocity V is in the + x direction. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 58

Example 9 -12 (b) n What is the fractional loss in kinetic energy? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 59

9 -10 Elastic Collisions in One Dimension n In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 60

9 -10 Elastic Collisions in One Dimension 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 61

9 -10 Elastic Collisions in One Dimension n Let us look at a few special situations: 1. Equal masses: If m 1 = m 2, and : Pool player’s result 2. A massive target: If m 2 >> m 1, and 3. A massive projectile: If m 1 >> m 2, and 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 62

Example 9 -13 n Two metal spheres, suspended by vertical cords, initially just touch, Sphere 1, with mass m 1 = 30 g, is pulled to the left to height h 1 = 8. 0 cm, and then released from rest. After swinging down, it undergoes an elastic collision with sphere 2, whose mass m 2 = 75 g. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 63

Example 9 -13 What is the velocity v 1 f of sphere 1 just after the collision? n Solution: 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 64

9 -11 Collisions in Two Dimensions n Total n If linear momentum must be conserved. it is also elastic, and then n For a glancing collision (it is not head-on): (1) Along the x axis as 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 65

9 -11 Collisions in Two Dimensions (2) Along the y axis as (3) 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 66

9 -12 Systems with Varying Mass: A Rocket n Finding the Acceleration and Velocity: n In Fig. (a) and (b), we show the rocket at times t and t + dt. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 67

9 -12 Systems with Varying Mass: A Rocket U is the velocity of the ejected gases with respect to the initial reference frame in which we measure the rocket’s speed v. 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 68

9 -12 Systems with Varying Mass: A Rocket ( first rocket equation ) ( second rocket equation ) 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 69

Example 9 -14 n. A rocket whose initial mass Mi is 850 kg consumes fuel at the rate R = 2. 3 kg/s. The speed vrel of the exhaust gases relative to the rocket engine is 2800 m/s. (a) What thrust does the rocket engine provide? (b) What is the initial acceleration of the rocket? 2021/2/20 普通物理講義-9 / 國立彰化師範大學物理系/ 郭艷光教授 70