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## Adapted from Holt book on physics

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**Chapter 10**Simple harmonic motion Adapted from Holt book on physics Section 1 Simple Harmonic Motion Section 2 Springs Section 3 Pendulum Section 4 Review**Simple Harmonic Motion**Defn: any periodic motion that is the result of a restoring force that is proportional to displacement. Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = = 2f = 2/T**Section 1 Simple Harmonic Motion**Chapter 10 Simple Harmonic Motion**Section 2 Measuring Simple Harmonic Motion**Chapter 10 Amplitude in SHM • In SHM, the maximum displacement from equilibrium is defined as theamplitude of the vibration. • Apendulum’s amplitude = swing angle from the vertical (Dq= radians) • For amass-spring system,the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. (Dx = meters)**Section 2 Measuring Simple Harmonic Motion**Chapter 10 Measures of Simple Harmonic Motion**Section 2 Measuring Simple Harmonic Motion**Chapter 11 Amplitude of a pendulum**Section 1 Simple Harmonic Motion**Chapter 11 Simple Harmonic Motion**Section 1 Simple Harmonic Motion**Chapter 11 Hooke’s Law • One type ofperiodic motionis the motion of a mass attached to a spring. • The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0).**Section 1 Simple Harmonic Motion**Chapter 11 Hooke’s Law, continued At equilibrium: • The spring force and the mass’s acceleration become zero. • The speedreaches amaximum. At maximum displacement: • The spring forceand the mass’saccelerationreach amaximum. • The speedbecomeszero.**Section 1 Simple Harmonic Motion**Chapter 11 Hooke’s Law, continued • Measurements show that thespringforce, or restoring force,isdirectly proportionalto thedisplacementof the mass. • This relationship is known asHooke’s Law: Felastic = –kx spring force = –(spring constant displacement) • The quantitykis a positive constant called thespring constant.**Section 1 Simple Harmonic Motion**Chapter 11 Spring Constant**Section 1 Simple Harmonic Motion**Chapter 11 Sample Problem Hooke’s Law If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?**Section 1 Simple Harmonic Motion**Chapter 11 Sample Problem, continued • 1.Define • Given: • m = 0.55 kg • x = –2.0 cm = –0.20 m • g = 9.81 m/s2 • Diagram: • Unknown: • k = ?**Section 1 Simple Harmonic Motion**Chapter 11 Sample Problem, continued 2. Plan Choose an equation or situation: When the mass is attached to the spring,the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass. Fnet = 0 = Felastic + Fg Felastic = –kx Fg = –mg –kx – mg = 0**Section 1 Simple Harmonic Motion**Chapter 11 Sample Problem, continued 2. Plan, continued Rearrange the equation to isolate the unknown:**Section 1 Simple Harmonic Motion**Chapter 11 Sample Problem, continued 3. Calculate Substitute the values into the equation and solve: 4. Evaluate The value of k implies that 270 N of force is required to displace the spring 1 m.**Section 1 Simple Harmonic Motion**Chapter 11 The Simple Pendulum • A simple pendulumconsists of amass called a bob, which is attached to a fixed string. • At any displacement from equilibrium, theweightof the bob(Fg)can be resolved into two components. • The xcomponent (Fg,x = Fg sin q)is theonly force acting on the bobin the direction of its motion and thus is therestoring force. The forces acting on the bob at any point are the force exerted by the string and the gravitational force.**Section 1 Simple Harmonic Motion**Chapter 11 The Simple Pendulum, continued • The magnitude of the restoring force (Fg,x = Fg sin q)is proportional to sin q. • When themaximum angle of displacement q is relatively small(<15°), sin qis approximately equal toqin radians. • As a result,the restoring force is very nearly proportional to the displacement. • Thus, the pendulum’s motion is an excellent approximation of simple harmonic motion.**Section 1 Simple Harmonic Motion**Chapter 11 Restoring Force and Simple Pendulums**Pendulum**• For “small oscillation”, period does not depend on • mass • amplitude Demos: M,A,L dependence**CORRECT**Concept Question Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should: 1. Make the pendulum shorter 2. Make the pendulum longer**CORRECT**Concept Question A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be 1. greater than T 2. equal to T 3. less than T “Effective g” is larger when accelerating upward (you feel heavier)**CORRECT**Concept Question If the amplitude of the oscillation (same block and same spring) was doubled, how would the period of the oscillation change? (The period is the time it takes to make one complete oscillation) 1. The period of the oscillation would double.2. The period of the oscillation would be halved3. The period of the oscillation would stay the same x +2A t -2A**m**x x=0 Potential Energy of a Spring Where x is measured fromthe equilibrium position PES x 0**m**Same thing for a vertical spring: y y=0 Where y is measured fromthe equilibrium position PES y 0**Section 1 Simple Harmonic Motion**Chapter 11 Force and Energy in Simple Harmonic Motion**CORRECT**Concept Question In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. In which case is the maximum kinetic energy of the mass the biggest? 1. Case 12. Case 23. Same**Concept Question**PE = 0 KE = KEMAX PE = 1/2kx2KE = 0 same for both samefor both x=+A x=0 x=-A x=+A x=0 x=-A**Simple Harmonic Motion:Quick Review**x(t) = [A]cos(t) v(t) = -[A]sin(t) a(t) = -[A2]cos(t) x(t) = [A]sin(t) v(t) = [A]cos(t) a(t) = -[A2]sin(t) OR Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = = 2f = 2/T xmax = A vmax = A amax = A2**Review: Period of a Spring**For simple harmonic oscillator = 2f = 2/T For mass M on spring with spring constant k Demos: A,m,k dependence**Chapter 11**Standardized Test Prep Multiple Choice Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 1. In what direction does the restoring force act? A. to the left B. to the right C. to the left or to the right depending on whether the spring is stretched or compressed D. perpendicular to the motion of the mass**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant? F. –5.0 10–2 N/m H. 5.0 10–2 N/m G. –2.0 101 N/m J. 2.0 101 N/m**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant? F. –5.0 10–2 N/m H. 5.0 10–2 N/m G. –2.0 101 N/m J. 2.0 101 N/m**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 3. In what form is the energy in the system when the mass passes through the equilibrium point? A. elastic potential energy B. gravitational potential energy C. kinetic energy D. a combination of two or more of the above**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 3. In what form is the energy in the system when the mass passes through the equilibrium point? A. elastic potential energy B. gravitational potential energy C. kinetic energy D. a combination of two or more of the above**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 4. In what form is the energy in the system when the mass is at maximum displacement? F. elastic potential energy G. gravitational potential energy H. kinetic energy J. a combination of two or more of the above**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 4. In what form is the energy in the system when the mass is at maximum displacement? F. elastic potential energy G. gravitational potential energy H. kinetic energy J. a combination of two or more of the above**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 5. Which of the following does not affect the period of the mass-spring system? A. mass B. spring constant C. amplitude of vibration D. All of the above affect the period.**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 5. Which of the following does not affect the period of the mass-spring system? A. mass B. spring constant C. amplitude of vibration D. All of the above affect the period.**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation? F. 8p s H. p s G. 4p s J.p/2 s**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation? F. 8p s H. p s G. 4p s J.p/2 s**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. • 7. What is the restoring force in the pendulum? • A. the total weight of the bob • B. the component of the bob’s weight tangent to the motion of the bob • C. the component of the bob’s weight perpendicular to the motion of the bob • D. the elastic force of the stretched string**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. • 7. What is the restoring force in the pendulum? • A. the total weight of the bob • B. the component of the bob’s weight tangent to the motion of the bob • C. the component of the bob’s weight perpendicular to the motion of the bob • D. the elastic force of the stretched string**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 8. Which of the following does not affect the period of the pendulum? F. the length of the string G. the mass of the pendulum bob H. the free-fall acceleration at the pendulum’s location J. All of the above affect the period.**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 8. Which of the following does not affect the period of the pendulum? F. the length of the string G. the mass of the pendulum bob H. the free-fall acceleration at the pendulum’s location J. All of the above affect the period.**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 9. If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum? A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 9. If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum? A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 10. If the pendulum’s length is 2.00 m and ag = 9.80 m/s2, how many complete oscillations does the pendulum make in 5.00 min? F. 1.76 H. 106 G. 21.6 J. 239**Chapter 11**Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 10. If the pendulum’s length is 2.00 m and ag = 9.80 m/s2, how many complete oscillations does the pendulum make in 5.00 min? F. 1.76 H. 106 G. 21.6 J. 239