Homework 5-1

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Aerospace Engineering

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Apr 3, 2024

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3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 1/26 Homework 5-1 Due: 11:57pm on Tuesday, February 13, 2024 To understand how points are awarded, read the Grading Policy for this assignment. Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by . The impulse is a vector pointing in the same direction as the force vector. The units of are or . Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object's momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object's motion: . So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of . Part A Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: Correct We often visualize the impulse by drawing a graph showing the force versus time. For a constant net force such as that used in the previous part, the graph showing the magnitude of the force versus time will look like the one shown in . = 8.4
3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 2/26 Part B The magnitude of the net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph showing the magnitude of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the magnitude of the force decaying to zero. It will look like the graph in . For this graph, the length of the rectangle corresponds to the impulse. For this graph, the height of the rectangle corresponds to the impulse. For this graph, the area of the rectangle corresponds to the impulse. For this graph, the slope of the rectangle corresponds to the impulse.
3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 3/26 Part C If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: Correct When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as . Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem. Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the + x direction. The impulse on the ball caused by the bat will be in the ANSWER: maximum force area slope
3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 4/26 Correct Part E Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the + x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's x - component of velocity just after leaving the bat if the bat applies an impulse of to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: 1. The blocks can move in only one dimension, namely, along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. positive x direction. negative x direction. = -26
3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 5/26 At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. Part A Find , the x component of the total momentum of the system at time . Express your answer in terms of , , , and . ANSWER: Correct Part B Find the time derivative of the x component of the system's total momentum. Express your answer in terms of , , , and . Hint 1. Finding the derivative of momentum for one block Consider the momentum of block 1: . Take the derivative of this expression with respect to time, noting that velocity is a function of time, and mass is a constant: . Hint 2. The relationship between velocity and acceleration Recall the definition of acceleration as . ANSWER: Correct Why did we bother with all this math? The expression for the derivative of momentum that we just obtained will be useful in reaching our desired conclusion, if only for this very special case. Part C The quantity (mass times acceleration) is dimensionally equivalent to which of the following? ANSWER: = =
3/1/24, 11:40 PM Homework 5-1 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID= 0 6/26 Correct Part D Acceleration is due to which of the following physical quantities? ANSWER: Correct Part E Since we have assumed that the system composed of blocks 1 and 2 is closed, what could be the reason for the acceleration of block 1? Hint 1. Force and acceleration Since the system is closed, the only object that can affect block 1 is the other block in the system, block 2. ANSWER: momentum energy force acceleration inertia velocity speed energy momentum force
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