5B HW 4

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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 1/21 HW 4 Due: 11:59pm on Friday, November 17, 2023 To understand how points are awarded, read the Grading Policy for this assignment. Position, Velocity, and Acceleration of an Oscillator Learning Goal: To learn to find kinematic variables from a graph of position vs. time. The graph of the position of an oscillating object as a function of time is shown. Some of the questions ask you to determine ranges on the graph over which a statement is true. When answering these questions, choose the most complete answer. For example, if the answer "B to D" were correct, then "B to C" would technically also be correct--but you will only recieve credit for choosing the most complete answer. Part A Where on the graph is ? ANSWER: Correct Part B Where on the graph is ? A to B A to C C to D C to E B to D A to B and D to E
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 2/21 ANSWER: Correct Part C Where on the graph is ? ANSWER: Correct Part D Where on the graph is the velocity ? Hint 1. Finding instantaneous velocity Instantaneous velocity is the derivative of the position function with respect to time, . Thus, you can find the velocity at any time by calculating the slope of the vs. graph. When is the slope greater than 0 on this graph? ANSWER: A to B A to C C to D C to E B to D A to B and D to E A only C only E only A and C A and C and E B and D
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 3/21 Correct Part E Where on the graph is the velocity ? ANSWER: Correct Part F Where on the graph is the velocity ? Hint 1. How to tell if The velocity is zero when the slope of the x vs. t curve is zero: . ANSWER: A to B A to C C to D C to E B to D A to B and D to E A to B A to C C to D C to E B to D A to B and D to E
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 4/21 Correct Part G Where on the graph is the acceleration ? Hint 1. Finding acceleration Acceleration is the second derivative of the position function with respect to time: . This means that the sign of the acceleration is the same as the sign of the curvature of the x vs. t graph. The acceleration of a curve is negative for downward curvature and positive for upward curvature. Where is the curvature greater than 0? ANSWER: Correct Part H Where on the graph is the acceleration ? A only B only C only D only E only A and C A and C and E B and D A to B A to C C to D C to E B to D A to B and D to E
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 5/21 ANSWER: Correct Part I Where on the graph is the acceleration ? Hint 1. How to tell if The acceleration is zero at the inflection points of the x vs. t graph. Inflection points are where the curvature of the graph changes sign. ANSWER: Correct Good Vibes: Introduction to Oscillations A to B A to C C to D C to E B to D A to B and D to E A only B only C only D only E only A and C A and C and E B and D
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 6/21 Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion . There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion . The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium . There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right.
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 7/21 Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we've assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period . Usually, the period is denoted and is measured in seconds. The frequency , denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: unchanged. doubled. halved. = 10
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 8/21 Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: = 0.025 R only Q only both R and Q K and L K and M K and P L and N M and P
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 9/21 Correct Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? = 0.02 = 0.01 = 0.48
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 10/21 Express your answer in meters. ANSWER: Correct Problem 15.2 - Enhanced - with Feedback In taking your pulse, you count 69 heartbeats in 1 . Part A What is the period of your heart oscillations? Express your answer in seconds. ANSWER: Correct Part B What is their frequency? Express your answer in hertz. ANSWER: Correct Problem 15.6 Part A What is the amplitude of the oscillation shown in ? Express your answer in centimeters. = 0.36 = 0.87 = 1.2
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 11/21 ANSWER: Correct Part B What is the frequency of the oscillation shown in the figure? Express your answer in hertz. ANSWER: Correct Problem 15.7 We can model the motion of a bumblebee's wing as simple harmonic motion. A bee beats its wings 230 times per second, and the wing tip moves at a maximum speed of 2.5 . Part A What is the amplitude of the wing tip's motion? Express your answer with the appropriate units. ANSWER: = 10 = 0.50 = 1.7
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 12/21 Correct Problem 15.13 Part A An object in simple harmonic motion has an amplitude of 8.0 , a frequency of 0.25 , and a phase constant of . Choose the correct position graph showing two cycles of the motion. ANSWER:
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 13/21 Correct Problem 15.15 An object in simple harmonic motion has amplitude 9.0 and frequency 0.36 . At = 0 it has its most negative position. Part A Write the function that describes the object's position. ANSWER: Correct Problem 15.16 - Enhanced - with Hints and Feedback An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 5.60 left of the equilibrium position and moving to the right at 40.6 . Part A What is the phase constant? Find the value between and . Express your answer with the appropriate units. Hint 1. How to approach the problem Express position and velocity as a function of time and use these equations to find at = 0 . Pay attention to the signs of the variables. ANSWER: = -2.09
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 14/21 Correct Part B What is the phase at ? Express your answer with the appropriate units. Hint 1. How to approach the problem Express the phase as a function of time using the phase constant of the previous part and find the phase at . ANSWER: Correct Part C What is the phase at ? Express your answer with the appropriate units. Hint 1. How to approach the problem Express the phase as a function of time using the phase constant of part A and find the phase at . ANSWER: Correct Part D What is the phase at ? Express your answer with the appropriate units. Hint 1. How to approach the problem Express the phase as a function of time using the phase constant of part A and find the phase at . ANSWER: = 0 = 2.09
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 15/21 Correct Problem 15.18 A block attached to a spring with unknown spring constant oscillates with a period of 1.7 . Parts A to D are independent questions, each referring to the initial situation. Part A What is the period if the mass is doubled? Express your answer with the appropriate units. ANSWER: Correct Part B What is the period if the mass is halved? Express your answer with the appropriate units. ANSWER: Correct Part C What is the period if the amplitude is doubled? Express your answer with the appropriate units. ANSWER: Correct = 4.19 = 2.4 = 1.2 = 1.7
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 16/21 Part D What is the period if the spring constant is doubled? Express your answer with the appropriate units. ANSWER: Correct Problem 15.19 A 160 air-track glider is attached to a spring. The glider is pushed in 11.8 and released. A student with a stopwatch finds that 12.0 oscillations take 11.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: Correct Problem 15.20 - Enhanced - with Hints and Feedback A 185 mass attached to a horizontal spring oscillates at a frequency of 3.40 . At = 0 , the mass is at = 5.40 and has = -16.0 . Part A Determine the period. Express your answer with the appropriate units. Hint 1. How to approach the problem Recall how the period of oscillations is related to oscillation frequency. = 1.2 = 7.5
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 17/21 ANSWER: Correct Part B Determine the angular frequency. Express your answer with the appropriate units. Hint 1. How to approach the problem Recall how angular frequency is related to oscillation frequency. ANSWER: Correct Part C Determine the amplitude. Express your answer with the appropriate units. Hint 1. How to approach the problem Apply the law of conservation of energy to express the maximum potential energy of the spring during SHM and then find the amplitude. ANSWER: Correct Part D Determine the phase constant. Express your answer with the appropriate units. Hint 1. How to approach the problem = 0.294 = 21.4 = 5.45×10 −2
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 18/21 Express position as a function of time and use the equation to find the phase at = 0 . Pay attention to the signs of the variables. ANSWER: Correct Part E Determine the maximum speed. Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the relation between the amplitude, velocity, and angular frequency to find the maximum speed. ANSWER: Correct Part F Determine the maximum acceleration. Express your answer with the appropriate units. Hint 1. How to approach the problem Recall how to find the maximum acceleration if you know the angular frequency and the amplitude of SHM. ANSWER: Correct Part G Determine the total energy. Express your answer with the appropriate units. = 0.138 = 1.16 = 24.9
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11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 19/21 Hint 1. How to approach the problem Find the kinetic and potential energy of the mass on the spring. Alternatively, you can find the maximum potential or kinetic energy. Then determine the total energy. ANSWER: Correct Part H Determine the position at = 3.00 . Express your answer with the appropriate units. Hint 1. How to approach the problem Write the position equation of SHM using the results of the previous parts and use it to find the position at = 3.00 . ANSWER: Correct Problem 15.37 Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 . The period on Mars turns out to be 2.45 . Part A What is the Martian free-fall acceleration? Express your answer with the appropriate units. ANSWER: Correct = 0.125 = 9.56×10 −3 = 3.67
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 20/21 Problem 15.36 Part A The free-fall acceleration on the moon is 1.62 . What is the length of a pendulum whose period on the moon matches the period of a 2.00- -long pendulum on the earth? Express your answer in meters. ANSWER: Correct Problem 15.41 - Enhanced - with Feedback An elephant's legs have a reasonably uniform cross section from top to bottom, and they are quite long, pivoting high on the animal's body. When an elephant moves at a walk, it uses very little energy to bring its legs forward, simply allowing them to swing like pendulums. For fluid walking motion, this time should be half the time for a complete stride; as soon as the right leg finishes swinging forward, the elephant plants the right foot and begins swinging the left leg forward. Part A An elephant has legs that stretch 2.3 from its shoulders to the ground. How much time is required for one leg to swing forward after completing a stride? Express your answer with the appropriate units. ANSWER: Correct Here we learn how to determine the time required for one leg to swing forward after completing a stride. Part B What would you predict for this elephant's stride frequency? That is, how many steps per minute will the elephant take? Express your answer in steps per minute. ANSWER: = 0.331 = 1.2 = 97 steps/minute
11/20/23, 2:16 PM HW 4 https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10876062 21/21 Correct Here we learn how to determine how many steps the elephant makes per minute. Score Summary: Your score on this assignment is 100%. You received 14 out of a possible total of 14 points.
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