Chapter 14: Periodic motion

Transcription

1 Chapter 14: Periodic motion Describing oscillations Simple harmonic motion Energy of simple harmonic motion Applications of simple harmonic motion Simple pendulum & physical pendulum Damped oscillations Forced oscillations and resonance

2 What causes periodic motion? Equilibrium position Restoring Force

3 Equilibrium and Oscillation

4 Linear Restoring Forces & Simple Harmonic Motion

5 Some parameters used to describe periodic motion Amplitude: the maximum magnitude from equilibrium. Phase angle: the point in the cycle the motion when time is zero. Period: the time to complete one cycle. Frequency: the number of cycles per unit of time (1 Hz = 1 s -1 ). Angular frequency: the rate of change of an angular quantity. Angular frequency Frequency Period

6 Simple harmonic motion of spring/mass system The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation. where Equation of motion for Solution Parameters for harmonic motion Angular frequency: Frequency: Period:

7 What is the difference due to gravity between a mass oscillating on a vertically stretched, ideal spring and a mass oscillating on a horizontally stretched, ideal spring? 1. The vertical oscillation amplitude is larger due to gravity. 2. The frequency of oscillation is smaller due to gravity. 3. The vertical oscillation frequency varies as the square root of g. 4. The frequency of oscillation is larger due to gravity. 5. Gravity does not affect the oscillation frequency or amplitude.

8 Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a b c d B. d c b a C. a b c d

9 A mass m that is suspended from a spring with spring constant k, is pulled down to a point P that is a distance d below the equilibrium point and then released. Suppose it takes the mass t seconds to move from point P back to the equilibrium point. If instead, the mass were pulled a distance 2d below the equilibrium point, the time it would take to reach the equilibrium point when then released would be 1. twice as long times as long. 3. the same time. 4. 1/ 2 times as long. 5. half as long

10 Simple harmonic motion viewed as a projection Simple harmonic motion is the projection of uniform circular motion onto a diameter. From the diagram, we see that where. Therefore,.

11 The type of function that describes simple harmonic motion is A. linear. B. exponential. C. quadratic. D. sinusoidal. E. inverse.

12 Displacement, Velocity, and Acceleration in SHM We start with our expression for the displacement,. The phase can be found from. where the position at is given by. The amplitude can be found from.

13 Graphing simple harmonic motion The graph below shows the effect of different phase angles. The graphs below show x, v x, and a x for = π/3.

14 Behavior of v x and a x during one cycle The speed is maximum when in the equilibrium position and zero when the distance from the equilibrium position is at a maximum. The magnitude of the acceleration is maximum when the speed is zero and zero when the speed is greatest.

15 A mass attached to a spring oscillates back and forth as indicated in the position vs. time plot below. At point P, the mass has 1. positive velocity and positive acceleration. 2. positive velocity and negative acceleration. 3. negative velocity and positive acceleration. 4. negative velocity and negative acceleration.

16 Energy in SHM The total mechanical energy, E = K + U, is conserved in SHM.

17 Energy diagrams for SHM

18 An object hangs motionless from a spring. When the object is pulled down, the sum of the elastic potential energy of the spring and the gravitational potential energy of the object and Earth. (Be careful here!) 1. increases 2. stays the same 3. decreases

19 Angular SHM A coil spring exerts a restoring torque z =, where is called the torsion constant of the spring. The result is angular simple harmonic motion.

20 Vibrations of molecules Two atoms have centers a distance r apart, with the equilibrium point at r = R 0. The atoms have a restoring force given by the Leonard-Jones potential, If they are displaced a small distance x from equilibrium, x = r R 0, the restoring force is can be approximated as the restoring force from simple harmonic motion.

21 The simple pendulum A simple pendulum consists of a point mass (the bob) suspended by a massless, unstretchable string. where Also, for Therefore, where

22 A simple pendulum is pulled to the side and released. Its subsequent motion appears as follows: 1. At which of the above times is the displacement zero? C, G 2. At which of the above times is the velocity zero? A, E, I 3. At which of the above times is the acceleration zero? C, G 4. At which of the above times is the kinetic energy a maximum? C, G 5. At which of the above times is the potential energy a maximum? A, E, I 6. At which of the above times is kinetic energy being transformed to potential energy? D, H 7. At which of the above times is potential energy being transformed to kinetic energy? B, F

23 The physical pendulum A physical pendulum is any real pendulum that uses an extended body instead of a point-mass bob. For small amplitudes, its motion is simple harmonic. The equation of motion becomes with the angular frequency given by

24 A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, the person stands on the swing, the natural frequency of the swing is 1. greater. 2. the same. 3. smaller.

25 Real-world systems have some dissipative forces that decrease the amplitude. Damped oscillations The decrease in amplitude is called damping and the motion is called damped oscillation. where

26 Forced oscillations and resonance A forced oscillation occurs if a driving force acts on an oscillator. Resonance occurs if the frequency of the driving force is near the natural frequency of the system.

27 If you drive an oscillator, it will have the largest amplitude if you drive it at its frequency. A. special B. positive C. resonant D. damped E. pendulum