2 PendulumA pendulum is simply a mass (bob) suspended from a string that can swing back and forthThe time it takes for the pendulum to swing back and forth is called the period.Period depends on length of the pendulum, not on weight suspended.The back and forth motion is called simple harmonic motion (SHM)Produces a sine curve
3 The restoring force of a pendulum is a component of the bob’s weight x-component pushes the bob toward equilibriumAt small angles, a pendulum follows SHMAt large angles this breaks downPotential energy increases as displacement increasesCons of energy still applies PE + KE = const.
4 Hooke’s Law Hooke’s Law Felastic = -kx See Figure 12-1 k = spring constantx = displacement- = force is always opposite direction from displacementSee Figure 12-1A stretched or compressed spring has elastic potential energyThe following holds true for a pendulum or a springAt the equilibrium point, v is a maxAt max displacement, force and “a” are maxIn SHM restoring force is proportional to displacement
5 Sample Problem 12AIf a mass of .55kg attached to a vertical spring stretches the spring 2.0cm from it’s equilibrium position, what is the spring constant?Not accelerating so Nnet = 0Fnet = 0 = Felastic - FW
7 Pendulum Mass-SpringThe period of a simple pendulum can be calculated withperiod = 2Π x square root of length / gravityPeriod of a mass spring system in SHMperiod = 2Π x square root of mass / spring constant
8 ExWhen a mass of 25g is attached to a certain spring, it makes 20 complete vibrations in 4.0s. What is the spring constant?Calculate period first4.0s/20vibrations = .2s / vibrationNow use the formula
9 .2s = 2Π√(.025kg/k).1s/ Π = √(.025kg/k) square both sides.00101s2 = .025kg / kk = 25 N/m
10 Sample Problem 12BYou need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower?
11 Sample Problem 12CThe body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of s. For the first few seconds, the vibration approximates SHM. Find the spring constant of a single spring.
13 Wave MotionMechanical waves travel by molecules vibrating back and forth or up and down.Sound, ocean waveThey have to travel through a mediumWater, airPulse – single traveling wave
14 Wave TypesTransverse Wave - Motion of the wave is at right angles to the direction the wave is moving.Longitudinal Wave - Motion of the wave is in the same direction as the direction the wave is moving.
16 Wave Description Waves are made up of several parts Crests - high pointsTroughs - low pointsAmplitude - distance from the midpoint to the crest or troughWavelength - distance form the top of one crest to the top of the next.Frequency - How often a vibration occurs.Period - amount of time for 1 cycle
18 FrequencyFrequency measures the number of times a wave oscillates in a given time (second)Frequency is measured in hertz1 cycle per second = 1 hertzfrequency = 1/periodperiod = 1/frequency
19 Wave SpeedWave speed depends on what type of medium it is traveling throughThe speed of a wave is dependant upon 2 things, wavelength and frequencyWave speed = wavelength x frequencyv = f
20 ExamplesThe Sears tower swings back and forth at a frequency of .1Hz what is its period?What is the wavelength of a 170Hz sound wave when the speed of sound in air is 340m/s?Frequency = 1/period.1Hz = 1/periodPeriod = 10sv = f340m/s =(170Hz) = 2.0m
21 Sample Problem 12DThe piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.
22 Interference Occurs every time two waves overlap Constructive - When crests of two waves overlap, it results in increased amplitude. The waves are said to be in phaseDestructive - When the crest of one wave overlaps with the trough of another wave, they cancel each other out. The waves are said to be out of phase with each other.
24 ReflectionWhen ever a wave transfers from one medium to another, part of the wave reflects and part of the waves energy continues on.At a free boundary, waves are reflected.At a fixed boundary, waves are reflected and inverted.
25 Standing WavesIf a string attached to a wall vibrates at exactly the right frequency, it can produce a standing wave.Standing Wave- A wave pattern that does not move along the string.Node – There is no motion on the stringAntinode – midway between the nodes, vibrations have the largest amplitude
26 Review What are the two types of interference? How fast is a wave with a wavelength of 3m and a frequency of 212Hz moving?Describe the two types of waves.If a wave has a period of .2s, what is its frequency?A radio station has a frequency of 101 MHz, what is the period of its wave?What are the two points of importance on a standing wave?