CHAPTER 7: OSCILLATORY MOTION REQUIRES A SET OF CONDITIONS 7.1 Period and Frequency Anything that vibrates or repeats its motion regularly is said to have oscillatory motion (sometimes called harmonic motion.) Any oscillatory motion repeats in regular intervals of time (the period of the motion) and does so at a constant rate (the frequency of the motion.) The relationship between period (T, measured in seconds) and frequency (f, measured in vibrations per second or Hertz) is given by: T = 1 f or f = 1 T. EXERCISE 7.1 1. A bee flaps his wings 700 times in 5.00 seconds. What is the frequency of the sound produced? 2. Waves move a fishing boat up and down 8.0 times in 11 seconds. What is the period of this motion? How many times will the boat move in 15 seconds? 7.2 Simple Harmonic Motion The characteristics of oscillatory systems can be determined by analyzing the motions of two such systems, a mass moving horizontally to stretch and compress a spring, and a simple pendulum swinging through small arcs. Each of these systems (and many others) show simple harmonic motion, or SHM. The characteristics of SHM can be seen in a horizontal spring-mass system (text p. 354, and on the following page.) In such a system, a mass slides on a horizontal, frictionless surface; the mass is attached to a spring and the other end of the spring is held stationary. The forces produced by a stretched or compressed spring are described by Hooke s Law. EXERCISE 7.2 1. State Hooke s Law for springs and other spring-like object (page 349.) Also give the equation (with units) for Hooke s Law. 2. Paraphrase Hooke s Law for a spring, along the lines of: The more a spring is stretched or compressed,.. 3. Sketch the graph of force-displacement for a Hooke s Law spring. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 1
4. Describe how some (unlikely) building materials (page 351) obey Hooke s Law. What is the major limitation to how well this works? 5. What is the meaning of restoring force for a stretched or compressed spring? As noted above, a horizontal spring-mass system provides a good illustration of simple harmonic motion. Study your text pages 354-355 and the following page. Note that this is a conservative system energy losses due to friction are ignored. With no forces applied, the spring is not stretched or compressed, and exerts no force on the mass; the mass remains in a centred or equilibrium position (remember the meaning of equilibrium: no net force is acting.) The mass is then pulled (by an external net force) sideways to the right so as to stretch the spring to its maximum, and then released. Stretching the spring stores elastic potential energy; when the mass is released, the E s is converted to E k as the mass moves towards the equilibrium position. This occurs because the stretched spring exerts a restoring force on the mass; the restoring force returns the mass to its equilibrium position. At the equilibrium position, the moving mass has converted all of the E s to E k, and continues to move through the equilibrium position. This E k is now converted progressively back into E s as the spring compresses. When the spring is finally compressed as much as it was initially stretched (away from the equilibrium position), all the E k has been converted to E s. The compressed spring exerts a restoring force again on the mass, accelerating it back towards the equilibrium position, and the whole harmonic motion begins again. 6. On the diagram on the following page, provide the required information (as indicated) for each position of the moving mass as it oscillates from side-to-side. Your answers will be: +max, max, or 0. Positive is to the right; be careful of the signs. Note that all forces and motions are horizontal; gravity is balanced by a normal force from the horizontal surface. Assume the system is conservative. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 2
Negative displaement Equilibrium position Positive displacement a) Equilibrium - no net force applied x +x b) x = 0 Applied positive force stretches spring F c) Applied force removed d) e) f) g) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 3
7. What name is given to an object that is moving with SHM? 8. A vertical spring-mass system (text p. 357) behaves in a manner similar to the horizontal one described above. The major difference is in the location of the equilibrium position. Which sketch on page 356 shows the system in equilibrium? How is the spring different at this position than for the corresponding position in the horizontal system? 9. Name the three examples of systems that vibrate with SHM, as given on pages 358 and 359. 10. Summarize the characteristics of SHM as stated in your text on page 358. Approximate SHM for a Pendulum A simple pendulum, for small amplitudes of swing, also shows approximate SHM. Once the pendulum has been pulled off to one side and released, the restoring force is provided by a component of gravity. Although this force acts tangentially to the circle of swing, for small swings (less than about 15 ), it is approximately horizontal. The sketches at right show the forces acting on a pendulum; the restoring force has a magnitude of F g sin θ. Your text (p. 377) shows how making this restoring force equal to that for a spring-mass system gives an equation for the period of a pendulum. F R F T F g θ F g θ F T F R PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 4
11. A pendulum swinging towards equilibrium is accelerated by the restoring force F g sin θ. If the pendulum has mass m, use Newton s second law to find an equation for its acceleration. 12. A pendulum of mass 250 g is pulled to an angle of 9.50 from vertical and released. What is the restoring force? What is the pendulum s initial acceleration in the direction of the restoring force? (0.405 N ; 1.62 m/s 2 ) 13. The sketch at right shows a 1.50 kg mass in a horizontal mass-spring system. The mass has been pulled 15.0 cm, to the right, away from the equilibrium position; the spring has a constant of 10.0 N/m. a) What is the restoring force (provided by the spring) at this position? Equilibrium position 1.50 kg x = 15.0 cm (1.50 N) b) What is the initial acceleration of the mass after it is released? (1.00 m/s 2 ) c) How much potential energy is stored in the spring at this position? (0.113 J) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 5
d) This is a conservative (frictionless) system; mechanical energy remains constant. This means that the potential energy of the mass at a displacement of 15.0 cm should equal the kinetic energy of the mass as it returns to equilibrium. Find how fast the mass will be traveling as it passes through the equilibrium position. (0.387 m/s) e) How fast will the mass be traveling as it passes a point 8.00 cm from equilibrium? (Use energy conservation. Remember that the total mechanical energy is constant, and equal to your answer for part (c). Find the spring potential energy at 8.00 cm; subtract this value from the constant total mechanical energy. This remaining energy must be kinetic, and you can find the speed of the mass at this point.) (0.340 m/s) 7.3 Position, Velocity, Acceleration and Time Relationships EXERCISE 7.3 1. State the equation (with units) for the period of a simple harmonic oscillator (Given on page 374 you don t need to know how to do this derivation.) 2. A 500 g mass is used in a simple harmonic oscillator with a spring with k = 65.0 N/m. What is the period of oscillation? (0.551 s) 3. A harmonic oscillator using a spring has a frequency of 2.50 Hz. If the mass used is 1.20 kg, what is the spring constant? (296 N/m) 4. On pages 378-379, your text derives the equation for the period of a pendulum by using the harmonic oscillator equation. (You don t need to know how to do this derivation.) What (probably surprising) value does not appear in this equation? Can you see a parallel in how the acceleration of a freely falling object does not depend on the mass of the object? Explain. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 6
5. State the equation for the period of a pendulum; include the units for the variables. 6. Rearrange the pendulum equation to solve for the length, l; to solve for the value of a g. How can a pendulum be used to experimentally determine the gravitational field strength at any point on earth (or any other planet or moon)? 7. Find the frequency of a pendulum with a length of 45.0 cm. 8. What length is needed for a pendulum with a period of 2.50 s? (0.743 Hz) (1.55 m) 9. A 120 cm pendulum operated on the moon takes 2.72 s to swing from left to right (not over and back.) Find the value of a g on the moon given by this information. (1.60 m/s 2 ) 10. A 1.6 kg mass on the end of a 1.1 m long string is used as a pendulum. How much more time will the pendulum need to make ten complete swings if the string is lengthened by 50 cm? (4.34 s) 11. Find the acceleration of gravity on a planet where a 45 cm long pendulum has a frequency of 0.50 Hz. (4.44 m/s 2 ) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 7
7.4 Applications of Simple Harmonic Motion EXERCISE 7.4 1. Define resonant frequency. How would you explain the use of natural in this definition? How might you demonstrate the resonant frequency of a vertical spring-mass system? Of a metre stick? Of a glass or beaker partially full of water? 2. When energy is added to a vibrating or oscillating system, the amplitude of vibration or oscillation may or may not increase. Under what conditions would amplitude increase? How often would energy need to be added to a pendulum with a length of 56.0 cm to increase the amplitude of the pendulum s swing? How often would this have to be done for a mechanical spring-mass oscillator with a constant of 59.0 N/m and a mass of 60.0 g? (every 1.50 s; every 0.200 s) 3. Define mechanical resonance as given in the text. See if you can write a better definition using the idea of efficiency in the conversion of input energy into output vibrations. 4. Describe ways in which buildings and bridges are designed to reduce the effects of wind-driven resonance. 5. As you saw in the Harmonic Motion Video, another source of energy exists that can cause resonance in buildings. What is this source, and how can it be protected against? 6. Mechanical clocks keep time using the regular oscillations of a pendulum. How do most electronic watches and clocks keep time? PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 7 REVISED JANUARY 08 PAGE 8