Wave Motion: v=λf [m/s=m 1/s] Example 1: A person on a pier observes a set of incoming waves that have a sinusoidal form with a distance of 1.

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1 Wave Motion: v=λf [m/s=m 1/s] Example 1: A person on a pier observes a set of incoming waves that have a sinusoidal form with a distance of 1.6 m between the crests. If a wave laps against the pier every 4.0 s, what are (a) the frequency and (b) the speed of the waves?

2 Standing Waves and Resonance: Example 2: A piano string with a length of 1.15 m and a mass of 20.0 g is under a tension of 6300 N. (a) What is the fundamental frequency of the string when it is struck? (b) What are the frequencies of the next two harmonics?

3 Example 3: Suppose you want to increase the fundamental frequency of a guitar string. (a) Would you (1) loosen the string to halve its tension, (2) tighten the string to double its tension, (3) use another string of the same material with half the diameter at the same tension, or (4) use another string of the same material with twice the diameter at the same tension? (b) In actuality, you want to go from the A note (220 Hz) below middle C to the A note (440 Hz) above middle C. IF the guitar strings are made of steel (ρ=7800 kg/m^3, Table 9.2), and an initial thicker string has a diameter of 0.30 cm, show that the string with half the diameter will have double the fundamental frequency.

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5 Simple Harmonic Motion: F = -kx (Hooke's Law) f=1/t PE = (1/2)kx Simple Harmonic Motion is motion under the influence of the type of force described by Hooke's Law because the force is the simplest restoring force and because the motion can be described by harmonic functions (sines and cosines). displacement: the directed distance of an object from its equilibrium position Amplitude (A): The magnitude of the maximum displacement, or the maximum distance, of an object from its equilibrium position period (T): The time for one complete cycle of motion frequency (f): The number of cycles per second (in Hz) or inverse seconds, where f=1/t)

6 Energy and Speed of a Mass-Spring System in SHM:

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9 A block with a mass of 0.25 kg sitting on a frictionless surface is connected to a light spring that has a spring constant of 180 N/m. If the block is displaced 15 cm from its equilibrium position and released, that are (a) the total energy of the system and (b) the speed of the block when it is 10 cm from its equilibrium position?

10 When a 0.5 kg mass is suspended from a spring, the spring stretches a distance of 10 cm to a new equilibrium position. (a) What is the spring constant of the spring? (b) The mass is then pulled down another 5 cm and released. What is the highest position of the oscillating mass?

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13 Wednesday, April 15 (Tax Day) - Take out your HW from Spring break - Any CH 13?s

14 Section 13.2: Equations of Motion (Our LAST Lesson before the AP Exam!!!) Equation of Motion: The equation that gives the object's position as a function of time. On a circle, the y-coordinate of the object is given by: y = A sinθ If θ = ωt then y = A sin(ωt), but this only works if the object starts at y=0 If we don't start at y=0, then we need new equations: If we are initially at y=+a, then y = A cos (ωt) ω = 2πf f=1/t ω = 2π/T start at y=0 then y = A sin(2πft) or y = A sin(2πt/t) Period of an object oscillating on a spring: Frequency of mass oscillating on a spring:

15 Angular frequency of mass oscillating on a spring: Period of a simple pendulum: Example: A mass on a spring oscillates vertically with an amplitude of 15 cm, a frequency of 0.20 Hz, and an equation of motion of y = A sin (ωt), with y = 0 at t = 0 and initial upward motion. (a) What are the position and direction of motion of the mass at t=3.1 s? (b) How many oscillations (cycles) does the mass make in a time of 12 s?

16 A helpful older brother takes his sister to play on the swings in the park. He pushes her from behind on each return. Assuming that the swing behaves as a simple pendulum with a length of 2.50 m, (a) what would be the frequency of the oscillations, and (b) what would be the interval between the brother's pushes? Hello class, your test is on Monday! Your test is on Pendulums, Oscillations, Waves, and Sound :-) Please spend 15 minutes finishing up the problems from yesterday (# skip 43 & 45) Also, come up with questions that you have/need to go over about any of the above topics/old homework?s.