Unit 7: Oscillations

Transcription

1 Text: Chapter 15 Unit 7: Oscillations NAME: Problems (p ) #1: 1, 7, 13, 17, 24, 26, 28, 32, 35 (simple harmonic motion, springs) #2: 45, 46, 49, 51, 75 (pendulums) Vocabulary: simple harmonic motion, hertz, amplitude, phase, angular frequency, period, simple pendulum, physical pendulum, damped harmonic motion, forced harmonic motion, resonance, natural frequency, equations of motion Math: definitions: x = Asin(!t + ") f = 1 T derived formulas: x =!" 2 x " m k L g skills: I rmg no new math skills Key Objectives: derive the equation of motion for simple harmonic motion. apply Newton s Second Law to a variety of situations, solve for the equations of motion, and determine the period of motion, if it is simple harmonic motion. derive the formulas listed above. correctly use the equations above in a variety of word problems. identify, define and give examples for the vocabulary listed above. understand and explain the assumptions and approximations made in the above formulas. understand and explain what happens to the energy of an oscillating system. set up, but not solve, the equations of motion for a damped harmonic oscillator. explain qualitatively what happens to a damped harmonic oscillator, e.g. energies, amplitudes, periods, velocities, etc

2 Formulas:! = d" dt! = d" dt s = r! v = r! a t = r! a c = r! 2! =! i +! f 2! f ="t +! i! = 1 2 "t 2 +# i t +! i I =! mr 2 = " r 2 dm! =! r "! F! " = I#! L =! r!! p L = I! K = 1 2 I! 2 W =!d" # Other formulas you may want: K = 1 2 mv 2 U = mgh F c = mv 2 r Some Moments of Inertia you may want (M = mass, R = radius, L = length): Disk : 1 2 MR2 Hoop: MR 2 Hollow Spherical Shell: 2 3 MR2 Solid Sphere: 2 5 MR2 Thin Rod, about center of mass: 1 12 ML2 Thin Rod, about one end: 1 3 ML2 Equations: x = x m sin(!t +") x =!" 2 x " f = 1 T m k L g I rmg U = 1 2 kx2 U = mgh K = 1 2 mv 2 F =!kx

3 Simple Harmonic Motion NAME: A pendulum swinging back and forth or a mass oscillating on a spring are two examples of Simple Harmonic Motion (SHM.) SHM occurs any time the position of an object as a function of time can be represented by a sine wave. We would then write the position as a function of time as x = Asin(!t + ") where t it time, A is the amplitude of the oscillations,! is called the phase shift of the motion, and is simply a constant that indicates the initial position of the object. Angles are measured in radians. The variable " is called the angular frequency. This can be seen by realizing that the sine function repeats every 2#. Calling the period of oscillation T, we can write (!(t + T) +") = (!t +") + 2# which means that!t = 2# " The velocity and acceleration of the object are found by taking the first and second derivative of the position: v =!Acos(!t + ") a = #! 2 Asin(!t + ") These are often written in dot form as x =!Acos(!t + ") x = #! 2 Asin(!t + ") Notice that the acceleration function is simply the position function multiplied by a constant, so that we can write x =!" 2 x This can be generalized so that anytime the acceleration is proportional to the position, the object is undergoing simple harmonic motion with period T and angular frequency ", as given above. Notice in the expressions above that we can also say the following: Displacement max = A Example 1: Mass on a spring Speed max = A! Acceleration max = A! 2 Consider a mass m on a horizontal frictionless table attached to a spring with a spring constant k. m k The spring can both stretch and contract, and the position x is defined to be zero at the unstretched position shown. Since we are assuming a frictionless table, the only force on the spring would be because of any stretching that occurs in the spring, according to Hooke s Law, so that side 1

4 Simple Harmonic Motion F =!kx ma =!kx m x =!kx x =! k m x NAME: This is the equation for simple harmonic motion derived earlier, so the angular frequency of the object on the spring is! 2 = k m! = The period of the object on the spring is related to the angular frequency so " k m m k This equation is also valid for a mass hanging from a spring. Example 2: A Simple Pendulum Consider a mass m hanging from pendulum of length L. To find an expression for the period of this pendulum, we will go through an analysis like we did for the spring: apply Newton s Laws, look for the simple harmonic relationship to find the angular frequency and solve for the period. Below is a diagram of the pendulum and the free-body diagram for the mass. T L mgsin m mg There are only two forces acting on the mass as it falls, the tension in the string and the weight of the mass. The net force on the mass varies with time, and always has two components to it. The net force component that is parallel to the string is the instantaneous centripetal acceleration of the mass, and the component that is perpendicular to the string is the instantaneous linear acceleration. Note that the centripetal force is greatest at the bottom of the swing when the linear acceleration is zero (no horizontal component of weight), and the linear acceleration is greatest at the top of its swing, when the centripetal force is zero (v=0). The position of the mass would be given by the angle $ alone, since we are assuming the string does not stretch in this process, and so we can ignore the centripetal component of the net force. The linear force on the mass is thus side 2

5 Simple Harmonic Motion NAME: F =!mg sin" ma =!mgsin" a =!gsin" The minus sign is because the direction of the acceleration is opposite the direction of the position. The linear acceleration is related to the angular acceleration so that a = L! = "gsin# which can be rewritten as! = " g L sin! Now we make an approximation. For small angles, sin$ % $ (remember that this is in radians. This is a common approximation in physics. See Note 2 at end.) So we get! = " g L! which is the simple harmonic motion equation with The period of the pendulum is therefore! 2 = g L " L g This holds true for small angles. The period of a pendulum depends only on the length of the pendulum, for small angular displacements, and of course the acceleration of gravity. Example 3: The Physical Pendulum Any body that is hanging and swinging is a pendulum, but the equation derived above is only valid for simple pendulums, a mass swinging on a massless string. A physical pendulum simply refers to any rigid body that is swinging back and forth, and we will derive an expression for its period in the same manner as before, except that now we will be applying Newton s Laws in rotational form. We will again be making the small angle approximation, but the only other restriction is that the body is a rigid one (and we will see that the simple pendulum equation already derived is just a special case of the physical pendulum. We start off with a rigid body hanging from a point that is not its center of mass (CM). This will produce a torque on the body, causing it to swing back and forth. Note that if the body was hanging from the center of mass, there would be no net torque on the body, and it would not swing. r CM side 3

6 Simple Harmonic Motion NAME: The net torque on the body will be because of gravity acting on the center of mass so that! " = r # F = I$ rmg sin% = I$ Rewriting the equation in dot form, and making the small angle approximation gives us! = " rmg! I Which is the simple harmonic motion equation. Solving for the period, we then get I rmg where I is the moment of inertia, m the mass, g the acceleration due to gravity and r is the distance from the point of oscillation to the center of mass. In the case of the simple pendulum, r is the length of the pendulum, L, and I is ml 2. Substituting this into the equation just derived yields which is what we found in Example 2. ml2 Lmg = 2! L g Note 1: Vertical Spring Hanging a mass vertically has the exact same period of oscillation. To see this, consider a mass hanging from a spring. We will define y=0 to be the position of the mass when the spring is unstretched, y 0 to be the equilibrium position of the mass, and down to be positive. By definition, this equilibrium position is when the weight of the mass is balanced by the spring force, or mg = ky 0 Now if we displace the spring a little, there is a net force on the mass trying to return it to its equilibrium position, so that " F = ma = mg! ky m y = mg! ky This almost looks like the Simple Harmonic Equation, except for the constant weight term. This can be taken care of by changing the coordinates. Let s define x = y! y 0 so that the variable x is zero at the equilibrium position, and is positive when the mass is below this point, and negative when above it. From that definition, we can take the first and second derivatives so that x = y x = y Now substitute x s for all the y s in the equation above and doing a little algebra gives us m y = mg! ky Since mg = ky 0, this reduces to m x = mg! k( x + y 0 ) m x =!kx + mg! ky 0 side 4

7 Simple Harmonic Motion m x =!kx x =! k m x NAME: which is the Simple Harmonic Equation we derived earlier. It doesn t matter if a mass on a spring is hanging vertically, at an angle or is horizontal; its period of oscillation will be given by m k Note 2: Small Angle Approximation The small angle approximation tends to annoy people the first time they see it. To help see how valid it is, the chart below shows values of $ and sin$. for a few angles. The approximation works pretty well (< 5% error) for angles smaller than 30 degrees, and very well (< 1% error) for angles smaller than 10 degrees. $ (radians) sin $ % difference $ (degrees) side 5

8 Oscillation Problems I NAME: 1. The position as a function for a 0.5 kg mass on the end of a spring is given by x = 1.5cos(3t). a. What is the maximum displacement of the mass from the equilibrium position? b. What is the period of this motion? c. What is the maximum speed of the mass? 2. A mass on a spring has an angular frequency of 5 rad/s and a maximum speed of 3 m/s. a. What is its maximum displacement? b. What is its maximum acceleration? 3. A mass on a spring has a maximum speed of 1.5 m/s and a maximum displacement of 25 cm. What is the period of oscillation? 4. A mass on a spring is oscillating with a frequency of 20 rpm. It also has a maximum acceleration of 1.5 m/s 2. What is the amplitude of the oscillation? 5. The position as a function of time for an oscillating object is shown. What is the maximum speed of the object? x (m) 3 t (s) Answers: 1. a) 1.5 m b) 2/3 π s c) 4.5 m/s 2. a) 0.6 m b) 15 m/s 2 3) 1/3 π s 4) 0.34 m 5) 2π m/s side 1

9 Oscillation Problems II NAME: 1. A 150 gram mass at rest stretches a spring 6 cm when it is hanging from the spring. It is then pulled down an addition 3 cm and released. a. What is the spring constant? b. What is the period of the resulting oscillations? c. What is the maximum speed of the mass? 2. A 4 kg mass is attached to a spring with a spring constant of 350 N/m. It is oscillating with a maximum acceleration of 5 m/s 2. a. What is the period of the motion? b. What is the amplitude of the motion? c. What is the maximum speed of the motion? d. How much energy does the system have? 3. A 2.4 kg mass is attached to a spring on a frictionless hill with a base angle of 30º. The mass has a maximum speed of 1.5 m/s and the amplitude of the simple harmonic motion is 25 cm. a. What is the period of the motion? b. What is the spring constant? side 1

10 Oscillation Problems II NAME: 4. What is the equation of motion for a 300 gram object oscillating on the end of s spring with a spring constant of 500 N/m and a maximum speed of 2.3 m/s? 5. A mass oscillating on a spring has a total energy of 5 J, a maximum acceleration of 12 m/s 2 and a frequency of 3 Hz. What is the mass? 6. The position as a function of time for a 150 gram object attached to a spring is shown in the diagram below. What is the spring constant? x (cm) 5 t (s) Derive an expression for the period of oscillation for the system shown. The mass is on a horizontal frictionless surface, and between two springs of spring constants k 1 and k 2. k 1 k 2 m Answers: 1. a) 25 N/m b) 0.49 s c) 0.39 m/s 2.a) 0.67 m/s b) m c) 0.53 m/s d) 0.57 J 3.a) 1.05 s b) 86.4 N/m 4) x=(0.056)cos(40.8 t) 5) 60.9 kg d) 0.14 N/m 7) m k 1 + k 2 side 2

11 Oscillation Problems III NAME: 1. A simple pendulum has a period of 1.5 seconds on the earth. If the same pendulum on another planet has a period of 3 seconds, what is the acceleration due to gravity on that other planet? 2. The angular position as a function of time for a simple pendulum is shown below. 0.1 ø (rad) t (s) -0.1 a. What is the length of the pendulum? b. What is the maximum speed of the simple pendulum? 3. A simple pendulum of mass m and length L is hanging from a point "O." Directly underneath "O" is a pin "P" that is fixed in place. When the pendulum is released, the pin P becomes the new oscillation axis. P is L/2 beneath O. What is the resulting period of small oscillations? O L L/2 m P side 1

12 Oscillation Problems III NAME: 4. A physical pendulum is any body that is hung from a point (not its center of mass) and set oscillating back and forth. Calling the mass of the body "m" and the distance between the oscillation axis and the center of mass "r" and its moment of inertia about that axis "I", what is the period of small oscillation for a physical pendulum? 5. What is the period of small oscillation for a thin rod of mass M and length L that is oscillating about one of its end points? 6. A thin rod of mass 400 grams and length 75 cm is suspended from one of its ends. At its other end is a small spring (k = 125 N/m) attached horizontally to a wall. The system is in equilibirum when it is hanging vertically. What is the period of small oscillation? axis of rotation wall uniform rod mass m length L k Answers: 1) 2.5 m/s 2 2.a) 4.05 m b) 0.31 m/s 3)!(1+ 2) 6) 0.56 s L 2g 4) 2! I rmg 5) 2! 2L 3g side 2

13 Lab 15-1 : Springs NAME Purpose: 1. To determine the relationship between force and stretch in a spring. 2. To determine the relationship between mass and period for an oscillating mass on a spring. Materials: 1 Spring 1 hanger 1 Mass Set 1 StopWatch Procedure: l. Measure the mass of the hanger and record. Then attach the hanger to the spring and suspend it from a stand and clamp. 2. As best you can, determine the initial position of the hanger as the position of the hanger just before its weight starts to stretch the spring. Measure all the stretches from this position. 3. Beginning with the empty hanger, record the amount that the spring is stretched with the hanger just sitting there. 4. Give the hanger a little push or displacement to get it oscillating up and down. Record the time for 10 oscillations. 5. Repeat steps 3 and 4 for several more trials, each time changing the hanging mass by adding some mass to the hanger. Try and add as much as you can without overstretching the spring. (Different springs behave differently, so you will have to judge your range of masses.) Analysis: l. Calculate the force in the spring for each trial when the system was at rest. 2. Calculate the period of oscillation for each trial when the system was not at rest. 3. Make two graphs: Force vs. Stretch and Period vs. Mass. 4. Even though both graphs will look linear at first glance, only one of them actually is linear. (Hint: make sure you see the origin of the graph.) After linearizing the non-straight graph, put regression lines through the two resulting graphs. 5. After checking with your teacher, print the graphs so that everyone has a copy. (Don t forget labels, units and titles.) Data: Mass of Hanger: kg Hanging Mass (kg) Stretch (m) Time 10 Cycles (s) Force (N) Period (s) side 1

14 Lab 15-1 : Springs NAME Conclusion: 1. What was the equation that related the force in the spring to the amount of stretch in the spring? 2. What is the physical significance of the slope of the F vs x graph? 3. What was the equation that related the period of oscillation to the mass hanging? 4. Compare the units of the two slopes of the two equations above. What do you notice? (You will have to simplify one of the slopes.) 5. In general, how should the slope of the second graph depend on the slope of the first graph? 6. Imagine you have a mass oscillating on a spring. What would happen to the period of oscillation if a. the mass were doubled? b. the spring constant were doubled? c. the amplitude were doubled? side 2

15 Lab 15-2 : Simple Pendulum NAME Purpose: 1. To determine which of the following three factors affects the period (T) of a pendulum: a) the mass of the bob b) the angle the pendulum is pulled back or c) the length of the pendulum. 2. To determine an equation from a straight line graph that relates that factor to the period, T. Procedure: Part I: Determining which factors affect the period. You must determine if mass, release angle, or length affect the period of a pendulum. One of these will have a dramatic effect on the period; the other two will not. l. Set up two pendulums using l00 gram masses. Hang them side by side from a bar set up between two stands. Make them the same length. Pull them back to angles that are quite different. Record the angles. Let the pendulums go, and time how long it takes each to swing back and forth 10 times. In either case, don't let the angle get larger than 45º. 2. Replace one of the 100 g masses with a 200 g mass, thus making one twice as massive as the other. Keep the lengths the same for each. Pull them back to the same angle and release them. Time how long it takes each to swing back and forth 10 times. Record in the data table. 3. Set up two pendulums and make one short and one long. Time how long it takes each to swing back and forth 10 times. Record in the data table. Part II: Determining the mathematical relationship for the period. 4. Now that you have found the factor that affects T you will vary that factor for seven trials. For each of these trials, record in the second data table the time for 10 back and forth swings, and also the value you are changing. 5. Graph the data plotting T vs. (the factor). If your graph does not come out to be a straight line, re-graph as necessary to make it so and then write the equation. Your original graph and the modified graph (if necessary) must be included with this lab. Data: Part I: Time for 10 cycles (s) Period (s) Angle 1 = Angle 2 = Mass 1 = Mass 2 = Length 1 = Length 2 = side 1

16 Lab 15-2 : Simple Pendulum NAME Part II: Time for 10 cycles (s) period, T (s) factor varied. Insert it here Conclusion: 1. What is your equation for the period of a pendulum? 2. Using a 100 gram mass, calculate how long the pendulum would have to be in order for its period to be l second? How long would it have to be if a 200 g mass was used? 3. Discuss the effect of air resistance on the period of a pendulum. side 2

17 Purpose: Diagram: Lab 15-3: SHM Energy Conservation NAME: 1. To analyze the energy transformations of a mass oscillating on a spring 2. To predict a variety of graphs concerning the motions and energies of an oscillating mass, and then confirm (or refute) those predictions spring mass stand with clamps motion detector Procedure: This lab is a series of predictions and testing. You will not be graded on the accuracy of your predictions, but please do your best to think about what will happen. Some of the questions may seem difficult at first; one of the points of the lab is to think deeply about the concept involved. After coming to an agreement as a group, use the computer to make the graphs. 1. Use a little tape to secure the mass to the hanger. Open up the file Mac/Applications/Logger Pro3/Experiments/Physics with Computers/17c Energy in SHM.xmbl. 2. Make sure that the mass on the spring is at rest, and place the motion detector directly under the mass. Make sure the motion detector is set to cart mode. Hit the zero button on the graph this will cause Logger Pro to call the equilibrium position of the mass x=0. 3. Pull the mass down a little bit and let it go, if it is oscillating nicely, click on record and let Logger Pro make position and velocity graphs. If the graphs look smooth, then you do not need to take any more data. 4. Set all four axes to Autoscale by double clicking on each graph, then choosing Axes Options. Mass on spring: Period of Oscillation: Part 1: Position, Velocity and Acceleration verses time of the oscillating mass. Also make note of the period of the oscillations and record above. Part 2: Velocity and Acceleration verses Position Part 3: Kinetic Energy verses time Part 4: Kinetic Energy verses position side 1

18 Lab 15-3: SHM Energy Conservation NAME: Part 5: Potential Energy verses time Part 6: Potential Energy verses position Part 7: Three graphs at once: Kinetic, Potential and Total Energy of Mass verses time Part 8: Three graphs at once: Kinetic, Potential and Total Energy of Mass verses position Conclusions: 1. From your measured period, calculate the spring constant for your spring. 2. Based on the printed out graphs, what are the equations that describe position, velocity and acceleration as a function of time? 3. Which of the graphs you made were not functions? Explain why they still made sense. 4. What forms of energy were not taken into account in your graphs of Total Energy. How important were they to your results? side 2

19 NAME: Post-Lab 15-3: Energy Conservation The following questions are based on Lab 15-3: Energy Conservation. They all deal with a mass oscillating on a spring. Position refers to the height of the mass above the motion detector. Positive values are taken to mean up. Part 1: Position, Velocity and Acceleration verses time. 1. Sketch velocity vs time and acceleration vs time for the following height verses time graph. height t 2. Sketch free-body diagrams representing the forces acting on the mass at its highest point, the middle, and the lowest point. Draw vectors of an appropriate length and indicate in which direction the net force points. Part 2: Velocity and Acceleration verses Position 3. Acceleration verses height should have looked like the following graph. Derive an expression for this function. a height 4. Velocity verses height was not a function. Sketch that graph and explain what it means. Part 4: KE and PE verses position 5. Compare and contrast the graph of Kinetic Energy verses time with velocity verses time. 6. Compare and contrast the graph of Potential Energy verses time with height verses time. (Do this for the gravitational PE only.) side: 1

20 NAME: Post-Lab 15-3: Energy Conservation Part 4a: KE and PE verses velocity 7. The graph of KE verses velocity looks like the following graph. What is the mathematical expression that describes this graph? KE velocity 8. Sketch and interpret the graph of (Gravitational) Potential Energy verses velocity. Part 5: Total Energy of Mass verses time 9. Total Energy was first defined as KE + PE, (where KE = 1/2 mv 2 and PE = mgh) That graph looked like the following graph. Why does it appear that energy is not conserved? Energy t 10. Sketch the graph of Energy verses time, taking into account other forms of energy. Explain your graph. 11. We ignored the energy dissipated by air drag and the rise in internal energy of the spring. Did that mess up the results at all? side: 2