## Chapter 14 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −− t1 x = 0 A k m Part B Find the velocity of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). t1 = v k m A t v(t) = Typesetting math: 100% Part D Find the acceleration of the block as a function of time. Express your answer in terms of some of all of the variables: , , , and . ANSWER: Part E Specify when the magnitude of the acceleration of the block reaches its maximum value. Consider the following options: only once during one a. period of motion, b. when the block’s speed is zero, c. when the block is in the equilibrium position, d. when the block’s displacement equals either or , e. when the block’s speed is at a maximum. Choose the most complete answer. You did not open hints for this part. ANSWER: a k m A t a(t) = A −A Typesetting math: 100% Part F Find the kinetic energy of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part G Find , the maximum kinetic energy of the block. Express your answer in terms of some or all of the variables: , , and . ANSWER: a only b only c only d only e only b and d c and e b and c a and e d and e K k m A t K(t) = Kmax k m A Typesetting math: 100% Part H The kinetic energy of the block reaches its maximum when which of the following occurs? You did not open hints for this part. ANSWER: Mass and Simple Harmonic Motion Conceptual Question The shaker cart, shown in the figure, is the latest extreme sport craze. You stand inside of a small cart attached to a heavy-duty spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you. Kmax = The displacement of the block is zero. The displacement of the block is . The acceleration of the block is at a maximum. The velocity of the block is zero. A Typesetting math: 100% At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground. Part A What effect does dropping the sandbag out of the cart at the equilibrium position have on the amplitude of your oscillation? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Instead of dropping the sandbag as you pass through equilibrium, you decide to drop the sandbag when the cart is at its maximum distance from equilibrium. Part C This question will be shown after you complete previous question(s). Part D It increases the amplitude. It decreases the amplitude. It has no effect on the amplitude. Typesetting math: 100% This question will be shown after you complete previous question(s). Simple Harmonic Motion Conceptual Question An object of mass is attached to a vertically oriented spring. The object is pulled a short distance below its equilibrium position and released from rest. Set the origin of the coordinate system at the equilibrium position of the object and choose upward as the positive direction. Assume air resistance is so small that it can be ignored. Refer to these graphs when answering the following questions. Part A Beginning the instant the object is released, select the graph that best matches the position vs. time graph for the object. You did not open hints for this part. ANSWER: m Typesetting math: 100% Part B Beginning the instant the object is released, select the graph that best matches the velocity vs. time graph for the object. You did not open hints for this part. ANSWER: Part C Beginning the instant the object is released, select the graph that best matches the acceleration vs. time graph for the object. A B C D E F G H A B C D E F G H Typesetting math: 100% You did not open hints for this part. ANSWER: Harmonic Oscillator Acceleration Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a function of time. One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass . The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be . The length of the relaxed spring is . The block is slowly pulled from its equilibrium position to some position along the x axis. At time , the block is released with zero initial velocity. The goal of this problem is to determine the acceleration of the block as a function of time in terms of , , and . It is known that a general solution for the position of a harmonic oscillator is , where , , and are constants. Your task, therefore, is to determine the values of , , and in terms of , ,and and then use the connection between and to find the acceleration. A B C D E F G H k m x = 0 L xinit > 0 t = 0 a(t) k m xinit x(t) = C cos (t) + S sin (t) C S C S k m xinit x(t) a(t) Typesetting math: 100% Part A Combine Newton’s 2nd law and Hooke’s law for a spring to find the acceleration of the block as a function of time. Express your answer in terms of , , and the coordinate of the block . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C a(t) k m x(t) a(t) = Typesetting math: 100% This question will be shown after you complete previous question(s). ± Introduction to Simple Harmonic Motion Consider the system shown in the figure. It consists of a block of mass attached to a spring of negligible mass and force constant . The block is free to move on a frictionless horizontal surface, while the left end of the spring is held fixed. When the spring is neither compressed nor stretched, the block is in equilibrium. If the spring is stretched, the block is displaced to the right and when it is released, a force acts on it to pull it back toward equilibrium. By the time the block has returned to the equilibrium position, it has picked up some kinetic energy, so it overshoots, stopping somewhere on the other side, where it is again pulled back toward equilibrium. As a result, the block moves back and forth from one side of the equilibrium position to the other, undergoing oscillations. Since we are ignoring friction (a good approximation to many cases), the mechanical energy of the system is conserved and the oscillations repeat themselves over and over. The motion that we have just described is typical of most systems when they are displaced from equilibrium and experience a restoring force that tends to bring them back to their equilibrium position. The resulting oscillations take the name of periodic motion. An important example of periodic motion is simple harmonic motion (SHM) and we will use the mass-spring system described here to introduce some of its properties. Part A Which of the following statements best describes the characteristic of the restoring force in the spring-mass system described in the introduction? You did not open hints for this part. ANSWER: m k The restoring force is constant. The restoring force is directly proportional to the displacement of the block. The restoring force is proportional to the mass of the block. The restoring force is maximum when the block is in the equilibrium position. Typesetting math: 100% Part B As shown in the figure, a coordinate system with the origin at the equilibrium position is chosen so that the x coordinate represents the displacement from the equilibrium position. (The positive direction is to the right.) What is the initial acceleration of the block, , when the block is released at a distance from its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: Part C What is the acceleration of the block when it passes through its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: a0 A A m k a0 = a1 A m k Typesetting math: 100% Part D This question will be shown after you complete previous question(s). Using the information found so far, select the correct phrases to complete the following statements. Part E You did not open hints for this part. ANSWER: Part F You did not open hints for this part. ANSWER: a1 = The magnitude of the block’s acceleration reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. Typesetting math: 100% Part G You did not open hints for this part. ANSWER: Part H Because of the periodic properties of SHM, the mathematical equations that describe this motion involve sine and cosine functions. For example, if the block is released at a distance from its equilibrium position, its displacement varies with time according to the equation , where is a constant characteristic of the system. If time is measured is seconds, must be expressed in radians per second so that the quantity is expressed in radians. Use this equation and the information you now have on the acceleration and speed of the block as it moves back and forth from one side of its equilibrium position to the other to determine the correct set of equations for the block’s x components of velocity and acceleration, and , respectively. In the expressions below, and are nonzero positive constants. You did not open hints for this part. The speed of the block is zero when it is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. The speed of the block reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between the rightmost position and the equilibrium position. between the leftmost position and the equilibrium position. A x t x = Acost t vx ax B C Typesetting math: 100% ANSWER: Period of a Pendulum Ranking Task Part A Six pendulums of mass and length as shown are released from rest at the same angle from vertical. Rank the pendulums according to the number of complete cycles of motion each pendulum goes through per minute. Rank from most to least complete cycles of motion per minute. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: , , , , vx = −Bsint ax = C cost vx = Bcost ax = C sint vx = −Bcost ax = −C cost vx = −Bsint ax = −C cost m L Typesetting math: 100% ± Gravity on Another Planet After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 47.0 . The explorer finds that the pendulum completes 108 full swing cycles in a time of 138 . Part A What is the magnitude of the gravitational acceleration on this planet? Express your answer in meters per second per second. You did not open hints for this part. cm s Typesetting math: 100% ANSWER: ± Tactics Box 14.1 Identifying and Analyzing Simple Harmonic Motion Learning Goal: To practice Tactics Box 14.1 Identifying and analyzing simple harmonic motion. A complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations among them. This Tactics Box summarizes the essential information needed to solve oscillation problems of this type. TACTICS BOX 14.1 Identifying and analyzing simple harmonic motion If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion around the equilibrium 1. position. The position as a function of time is . The velocity as a function of time is . The maximum speed is . The equations are given here in terms of , but they can be written in terms of , or some other parameter if the situation calls for it. 2. 3. The amplitude and the phase constant are determined by the initial conditions through and . 4. The angular frequency (and hence the period ) depends on the physical properties of the situtaion. But does not depend on or . Mechanical energy is conserved. Thus .Energy conservation provides a relationship between position and velocity that is independent of time. 5. Part A The position of a 60 oscillating mass is given by , where is in seconds. Determine the velocity at . Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: gplanet = m/s2 x(t) = Acos(t + 0 ) vx(t) = −Asin(t + 0 ) vmax = A x y A 0 x0 = Acos 0 v0x = −Asin 0 T = 2/ A 0 1 m + k = k = m( 2 v2 x 1 2 x2 1 2 A2 1 2 vmax)2 g x(t) = (2.0 cm) cos(10t) t t = 0.40 s Typesetting math: 100% Part B Assume that the oscillating mass described in Part A is attached to a spring. What would the spring constant of this spring be? Express your answer in newtons per meter to two significant figures. You did not open hints for this part. ANSWER: Part C What is the total energy of the mass described in the previous parts? Express your answer in joules to two significant figures. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. vx = m/s k k = N/m E E = J Typesetting math: 100%

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