Chapter 14 Oscillations
14-7 Damped Harmonic Motion Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an envelope that modifies the undamped oscillation. If then
14-7 Damped Harmonic Motion This gives If b is small, a solution of the form will work, with Try it; like we did in class yesterday.
14-7 Damped Harmonic Motion If b 2 > 4mk, ω becomes imaginary, and the system is overdamped (C). For b 2 = 4mk, the system is critically damped (B) this is the case in which the system reaches equilibrium in the shortest time.
14-7 Damped Harmonic Motion There are systems in which damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.
14-7 Damped Harmonic Motion A damped harmonic oscillator loses 6.0% of its mechanical energy per cycle. a) By what percentage does its frequency differ from the natural frequency f 0 = ( 1 2! ) k m?!f f 0 = 0.0012% b) After how many periods will the amplitude have decreased to 1/e of its original value? n =!1 ln( 0.94) " 16
14-8 Forced Oscillations; Resonance Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system. If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.
14-8 Forced Oscillations; Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp. Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.
14-8 Forced Oscillations; Resonance The equation of motion for a forced oscillator is: The solution after a a long time is: where and
14-8 Forced Oscillations; Resonance The width of the resonant peak can be characterized by the Q factor:
For SHM, the restoring force is proportional to the displacement. The period is the time required for one cycle, and the frequency is the number of cycles per second. Period for a mass on a spring: SHM is sinusoidal. Summary of Chapter 14 During SHM, the total energy is continually changing from kinetic to potential and back.
Summary of Chapter 14 A simple pendulum approximates SHM if its amplitude is not large. Its period in that case is: When friction is present, the motion is damped. If an oscillating force is applied to a SHO, its amplitude depends on how close to the natural frequency the driving frequency is. If it is close, the amplitude becomes quite large. This is called resonance.
Group question: A 2.00-kg mass oscillates on the end of a spring with spring constant 12.0 N/m. Its amplitude of oscillation decreases from 10.0 cm to 1.0 cm in 4.00 minutes. What is the linear damping coefficient of this oscillator? A) 134 N s/m B) 1.76 N s/m C) 0.311 N s/m D) 0.0384 N s/m E) 0.622 N s/m
Group question: A mass of 2.0 kg hangs from a spring with a force constant of 50 N/m. An oscillating force F = (4.8 N) cos[(3.0 rad/s)t] is applied to the mass. What is the amplitude of the resulting oscillations? Neglect damping. A) 0.15 m B) 0.30 m C) 1.6 m D) 2.4 m E) 0.80 m
Chapter 15 Wave Motion
Units of Chapter 15 Characteristics of Wave Motion Types of Waves: Transverse and Longitudinal Energy Transported by Waves Mathematical Representation of a Traveling Wave The Principle of Superposition Reflection and Transmission
Units of Chapter 15 Interference Standing Waves; Resonance Refraction Diffraction
15-1 Characteristics of Wave Motion All types of traveling waves transport energy. Study of a single wave pulse shows that it is begun with a vibration and is transmitted through internal forces in the medium. Continuous waves start with vibrations, too. If the vibration is SHM, then the wave will be sinusoidal.
15-1 Characteristics of Wave Motion Wave characteristics: Amplitude, A Wavelength, λ Frequency, f and period, T Wave velocity,
15-2 Types of Waves: Transverse and Longitudinal The motion of particles in a wave can be either perpendicular to the wave direction (transverse) or parallel to it (longitudinal).
15-2 Types of Waves: Transverse and Longitudinal Sound waves are longitudinal waves:
15-2 Types of Waves: Transverse and Longitudinal The velocity of a transverse wave on a cord is given by: As expected, the velocity increases when the tension increases, and decreases when the mass increases.
Assignment Chapter 14 60, 87 HAND IN FRIDAY