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10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 1/20 Chapter 08 Homework Assignment Due: 11:59pm on Thursday, October 31, 2019 You will receive no credit for items you complete after the assignment is due. Grading Policy Center of Mass and External Forces Learning Goal: Understand that, for many purposes, a system can be treated as a point-like particle with its mass concentrated at the center of mass. A complex system of objects, both point-like and extended ones, can often be treated as a point particle , located at the system's center of mass . Such an approach can greatly simplify problem solving. Before you use the center of mass approach, you should first understand the following terms: System: Any collection of objects that are of interest to you in a particular situation. In many problems, you have a certain freedom in choosing your system. Making a wise choice for the system is often the first step in solving the problem efficiently. Center of mass: The point that represents the "average" position of the entire mass of a system. The postion of the center of mass can be expressed in terms of the position vectors of the particles as . The x coordinate of the center of mass can be expressed in terms of the x coordinates of the particles as . Similarly, the y coordinate of the center of mass can be expressed. Internal force: Any force that results from an interaction between the objects inside your system. As we will show, the internal forces do not affect the motion of the system's center of mass. External force: Any force acting on an object inside your system that results from an interaction with an object outside your system. Consider a system of two blocks that have masses and . Assume that the blocks are point-like particles and are located along the x axis at the coordinates and as shown . In this problem, the blocks can only move along the x axis. Part A Find the x coordinate of the center of mass of the system. Express your answer in terms of , , , and . ANSWER:

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 2/20 Correct Part B If , then the center of mass is located: ANSWER: Correct Part C If , then the center of mass is located: ANSWER: Correct Part D Recall that the blocks can only move along the x axis. The x components of their velocities at a certain moment are and . Find the component of the velocity of the center of mass at that moment. Keep in mind that, in general: . Express your answer in terms of , , , and . ANSWER: = to the left of at a distance much greater than to the left of at a distance much less than to the right of at a distance much less than to the right of at a distance much greater than to the right of at a distance much less than to the left of at a distance much less than at at half-way between and the answer depends on and

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 4/20 Correct Part H Assume that the blocks are accelerating, and the x components of their accelerations at a certain moment are and . Find the x component of the acceleration of the center of mass at that moment. Keep in mind that, in general, . Express your answer in terms of , , , and . ANSWER: Correct Because and are the x components of the velocities of and their values can be positive or negative or equal to zero. We will now consider the effect of external and internal forces on the acceleration of the center of mass. Part I Consider the same system of two blocks. An external force is now acting on block . No forces are applied to block as shown . Find the x component of the acceleration of the center of mass of the system. Express your answer in terms of the x component of the force, ,and . Hint 1. Using Newton's laws Find the acceleration of each block from Newton's second law and then apply the formula for found earlier. none of the above =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 5/20 ANSWER: Correct Part J Consider the same system of two blocks. Now, there are two forces involved. An external force is acting on block and another external force is acting on block . Find the x component of the acceleration of the center of mass of the system. Express your answer in terms of the x components and of the forces, and . ANSWER: Correct Note that, in both cases, the acceleration of the center of mass can be found as where is the net external force applied to the system, and is the total mass of the system. Even though each force is only applied to one object, it affects the acceleration of the center of mass of the entire system . This result is especially useful since it can be applied to a general case, involving any number of objects moving in all directions and being acted upon by any number of external forces. Part K Consider the previous situation. Under what condition would the acceleration of the center of mass be zero? Keep in mind that and represent the components , of the corresponding forces. ANSWER: = =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 7/20 Part A Which of the following best describes this collision? Hint 1. Types of collisions An inelastic collision is a collision in which kinetic energy is not conserved. In a partially inelastic collision, kinetic energy is lost, but the objects colliding do not stick together. From this information, you can infer what completely inelastic and elastic collisions are. ANSWER: Correct Part B Which of the following quantities, if any, are conserved during this collision? Hint 1. When is kinetic energy conserved? Kinetic energy is conserved only in perfectly elastic collisions. ANSWER: perfectly elastic partially inelastic perfectly inelastic

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 8/20 Correct Part C What is the speed of the block/bullet system after the collision? Express your answer in terms of , , and . Hint 1. Find the momentum after the collision What is the total momentum of the block/bullet system after the collision? Express your answer in terms of and other given quantities. ANSWER: Hint 2. Use conservation of momentum The momentum of the block/bullet system is conserved. Therefore, the momentum before the collision is the same as the momentum after the collision. Find a second expression for , this time expressed as the total momentum of the system before the collision. Express your answer in terms of and other given quantities. ANSWER: ANSWER: Correct A Ball Hits a Wall Elastically A ball of mass moving with velocity strikes a vertical wall as shown in . The angle between the ball's initial velocity vector and the wall is as shown on the diagram, which depicts the situation as seen from above. The duration of the collision kinetic energy only momentum only kinetic energy and momentum neither momentum nor kinetic energy = = =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 10/20 ANSWER: Hint 4. Putting it together Once you find the vector components of the final velocity in terms of the initial velocity, use the geometry of similar triangles to determine in terms of . ANSWER: Correct Part B What is the magnitude of the average force exerted on the ball by the wall? Express your answer in terms of variables given in the problem introduction and/or . Hint 1. What physical principle to use Use the impulse-momentum theorem, , along with the definition of impulse, . In this case, only one force is acting, so . Putting everything together, . Hint 2. Change in momentum of the ball The fact that implies that the component of the ball's momentum does not change during the collision. What is , the magnitude of the change in the ball's momentum? Express your answer in terms of quantities given in the problem introduction and/or . ANSWER: ANSWER: Correct Exercise 8.12 = = = =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 11/20 A bat strikes a 0.145- baseball. Just before impact, the ball is traveling horizontally to the right at 60.0 , and it leaves the bat traveling to the left at an angle of 30 above horizontal with a speed of 65.0 . The ball and bat are in contact for 1.65 . Part A Find the horizontal component of the average force on the ball. Take the x -direction to be positive to the right Express your answer using two significant figures. ANSWER: Correct Part B Find the vertical component of the average force on the ball. Express your answer using two significant figures. ANSWER: Correct Exercise 8.17 The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A .30 caliber bullet has mass 7.20×10 −3 and a speed of 601 relative to the muzzle when fired from a rifle that has mass 3.00 . The loosely held rifle recoils at a speed of 1.95 relative to the earth. Part A Find the momentum of the propellant gases in a coordinate system attached to the earth as they leave the muzzle of the rifle. ANSWER: Correct Conservation of Momentum in Inelastic Collisions Learning Goal: To understand the vector nature of momentum in the case in which two objects collide and stick together. = −1.0×10 4 = 2900 = 1.54

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 13/20 Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, what is the magnitude of their combined momentum? Hint 1. How to approach the problem Think about how you would calculate the final momentum of the two cars using the information provided and the idea of conservation of momentum. Better yet, try the calculation out. What do you get? Keep in mind that momentum is a vector, but you are asked about the magnitude of the momentum, which is a scalar. ANSWER: Correct Part D Two cars collide inelastically and stick together after the collision. Before the collision, their momenta are and . After the collision, their combined momentum is . Of what can one be certain? Hint 1. Momentum is a vector Momentum is a vector quantity, and conservation of momentum holds for two-dimensional and three-dimensional collisions as well as for one-dimensional collisions. ANSWER: The answer depends on the directions in which the cars were moving before the collision.

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 14/20 Correct You can decompose the vector equation that states the conservation of momentum into individual equations for each of the orthogonal components of the vectors. Part E Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, the magnitude of their combined momentum is . Of what can one be certain? Hint 1. How to approach the problem mathematically Momentum is a vector quantity. It is impossible to make exact predictions about the direction of motion after a collision if nothing is known about the direction of motion before the collision. However, one can put some bounds on the values of the final momentum. Start with the expression for from Part D: . Therefore, , where is the angle between and . (To derive the above, you would have to break each vector into components.) So the value of is controlled by . Hint 2. How to approach the problem empirically Consider the directions for the initial momenta that will give the largest and smallest final momentum. ANSWER: Correct When the two cars collide, the magnitude of the final momentum will always be at most (a value attained if the cars were moving in the same direction before the collision) and at least (a value attained if the cars were moving in opposite directions before the collision). Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic. Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of . After the collision, the two cars stick together and travel off in the direction

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 16/20 ANSWER: Correct Part B Find the tangent of the angle . Express your answer in terms of the momenta of the two cars, and . ANSWER: Correct Part C Suppose that after the collision, ; in other words, is . This means that before the collision: ANSWER: Correct Exercise 8.42 A bullet of mass 5.00 is fired horizontally into a wooden block of mass 1.12 resting on a horizontal surface. The coefficient of kinetic friction between block and surface is 0.250. The bullet remains embedded in the block, which is observed to slide a distance 0.270 along the surface before stopping. Part A What was the initial speed of the bullet? = = = The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal.

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 17/20 ANSWER: Correct Trading Momenta in a Collision Two particles move perpendicular to each other until they collide. Particle 1 has mass and momentum of magnitude , and particle 2 has mass and momentum of magnitude . Note: Magnitudes are not drawn to scale in any of the figures. Part A Suppose that after the collision, the particles "trade" their momenta, as shown in the figure. That is, particle 1 now has magnitude of momentum , and particle 2 has magnitude of momentum ; furthermore, each particle is now moving in the direction in which the other had been moving. How much kinetic energy, , is lost in the collision? Express your answer in terms of and . Hint 1. How to approach the problem To find the kinetic energy lost in the collision, compute the initial kinetic energy (before the collision) and the final kinetic energy (after the collision). Then take the difference: . Hint 2. Find the relationship between energy and momentum What is the kinetic energy of a particle with mass and magnitude of momentum ? ANSWER: Hint 3. Find the initial kinetic energy What is the initial kinetic energy of the two-particle system? Express your answer in terms of and . = 259 =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 19/20 Hint 1. How to approach the problem As in Part A, to find the kinetic energy lost in the collision, compute the initial kinetic energy (before the collision) and the final kinetic energy (after the collision). Then take the difference: . Hint 2. Definition of completely inelastic A completely inelastic collision is one in which the particles stick together after the collision. Hint 3. Initial kinetic energy The initial kinetic energy in this case is the same as it was in the partially elastic collision. Hint 4. Find the final kinetic energy What is the combined kinetic energy of the two particles after the perfectly inelestic collision? If you use the formula , remember that is the magnitude of the momentum vector. Express your answer in terms of and . Hint 1. Find the magnitude of the final momentum What is the magnitude of , the total momentum of the two particles after the collision? Express in terms of . Hint 1. Find the final momentum vector What is the total momentum of the two particles after the collision? Take the positive x direction to be to the right in the figures and the positive y direction to be upward. Express your answer as a vector in terms of and the unit vectors and . ANSWER: ANSWER: ANSWER: ANSWER: = = =

10/27/2019 Chapter 08 Homework Assignment https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=7488219 20/20 Correct Problem 8.101 A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. Part A What fraction of the total energy released goes into the kinetic energy of the proton? Express your answer as a percentage. ANSWER: Correct Score Summary: Your score on this assignment is 101%. You received 10.14 out of a possible total of 10 points. = = 5.44×10 −2