Simple Harmonic Motion and Waves Simple Harmonic Motion (SHM) periodic motion that occurs whenever the restoring force is proportional to the displacement and in the opposite direction. Give some example of things that oscillate back and forth in a systematic way. For each of your examples determine what the restoring force could be. x a F v notes TELJR Publications 2011 1
The restoring force of an ideal spring is F s = -kx where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring. Since the force on the object is not constant, the acceleration of the object is not constant. Period and Frequency Period Frequency Q.1. On the diagrams below indicate what one cycle would look like. TELJR Publications 2011 2
Period of a Mass and Spring & Period of a Pendulum Mass and Spring Pendulum Factors: Factors: Relation: Relation: Formula: Formula: Q.1. Determine the period of a mass and spring system where the spring constant is 300 N/m and the mass is 0.20 kg. Q.2. How long would a pendulum string have to be to have the same period as Q.1.? TELJR Publications 2011 3
Q.3. Rewrite each of the above equations for frequency instead of period. Mass and Spring Pendulum TELJR Publications 2011 4
Energy and SHM Q.1. A block and mass system is shown to the right. The spring is then stretched to a displacement of A. a) What type of energy does the block have in diagram A? b) Energy is conserved, so what is true about the energy anywhere during the block s motion (refer to diagram (b))? c) When does the block have its greatest kinetic energy? Energy Energy -A A Energy vs. Displacement Energy vs. Time T 2 T TELJR Publications 2011 5
Traveling Waves There are two main types of waves, classified as mechanical and electromagnetic, with definitions of each given below. Based upon past lessons, generate examples of each type of wave. Mechanical Wave - Wave that requires a medium to transmit energy Examples: Electromagnetic Wave -Wave that does not require a medium to transmit energy Examples: Mechanical Waves can be broken down into two main families, transverse and longitudinal. Using the diagrams below, determine the relationship between the motion of the wave and the motion of the medium. Transverse Longitudinal Wave motion/medium motion: Wave motion/medium motion: TELJR Publications 2011 6
Use the diagram to the left to define the following terms: Amplitude: Crest: Trough: Wavelength: Displacement: Speed of a Wave Speed: v = d/t What is the distance of one cycle: What is the time for one cycle: v = = = Q.1. Determine the speed of a wave with a wavelength of 2.0 m and a period of 2.5 secs. Q.2. A fisherman notices waves passing under his boat every 3 seconds and roughly determines the speed to be 3.0 m/s. How far apart are the crests of the wave? TELJR Publications 2011 7
Different Types of Wave Speeds Waves on a String Sound Waves Electromagnetic Waves v = F T m/ L Speed of sound in air at STP: 330 m/s Speed of sound in air at room temp: 343 m/s Temperature dependence: Medium: As sound enters water from air: All EM waves travel at the same speed in a vacuum: the speed of light : c = 3.00 x 10 8 m/s In other media (air, glass, water), different EM waves slow down by different amounts. As light enters water from air: Examples: Examples: Examples: TELJR Publications 2011 8
Phase and Reflections In phase: (A,E,I) (B,F) (D,H) (C,G) Condition for points to be in phase: Completely out of phase: (A,C) (B,D) (A,G) (B,H) Condition for points to be completely out of phase: Soft Reflection Observations: Hard Reflection Observations: Conditions of Soft Reflections: Conditions of Hard Reflections: TELJR Publications 2011 9
Whole waves can also be in or out phase with each other, as shown below. IN PHASE OUT OF PHASE IN PHASE OUT OF PHASE Conclusions: TELJR Publications 2011 10
Principle of Superposition: Superposition and Interference Name the type of interference showing in the diagrams below: Observations: Observations: Name: Name: Examples: Examples: TELJR Publications 2011 11
Superposition and Standing Waves Standing waves are created when waves, traveling in opposite directions, constructively and destructively interfere with one another. In the diagrams to the left, where are the waves constructively interfering? Where are they destructively interfering? Determine the number of & in the diagrams to the left. Q.1. What is the distance, in terms of wavelength separating the following points: Node to next consecutive node Antinode to next consecutive antinode +Antinode to next consecutive +antinode What would be the length of two nodes and an antinode? TELJR Publications 2011 12
Applications of Standing Waves String Fixed at Both Ends Waves are produced on a string and have a wave speed of 1200 m/s 6 m 6 m 6 m 1 st Harmonic 2 nd Harmonic 3 rd Harmonic # of Nodes/Antinodes: # of Nodes/Antinodes: # of Nodes/Antinodes: Length (L) in terms of λ Length (L) in terms of λ Length (L) in terms of λ TELJR Publications 2011 13
Doppler Shift What is it? What does it mean for sound? Light? Wavefronts from a stationary source Observations: Wavefronts from a source moving to the right Observations: If the source and observers are stationary then.. If the source and/or the observers are moving relative to one another then. Getting Closer: #1 #2 Stationary source and stationary observers Moving source and stationary observers Getting further: TELJR Publications 2011 14
For truck moving at constant velocity: one constant high pitch when moving toward and one constant low pitch hear d when moving away For truck speeding up: pitch increases and then decreases RADAR How does a police car use the Doppler Effect to determine your speed? When looking at distance stars and galaxies, astronomers note whether or not the object is blue-shifted or red-shifted. What does that indicate about the object s velocity with respect to the Earth? Blue Shift Red Shift Conclusions: TELJR Publications 2011 15