Vibrations and Waves

Objectives: 1)Describe how energy and speed vary in simple harmonic motion2) Calculate sinusoidal wave properties in terms of amplitude, wavelength, frequency, and period.3) Identify different types of waves.

Bellwork 04/12/10

1. A paper airplane accumulates 0.80 x 10-12 C of charge while flying perpendicular to Earth’s magnetic field of 5.0 x 10-5 T. How fast is the paper airplane flying, neglecting air resistance, if the magnetic force acting on it is 6.4 x 10-16 N? **F=qvBsin**

2. What is the direction of the force on a positive charge that travels through a magnetic field as shown below?

S N

Bellwork Solutions

1. 6.4 x 10-16N=(0.8x10-12C)(v)(5.0 x 10-5T)(sin90o)v = 6.4 x 10-16 = 16 m/s (0.8x10-12)(5.0x10-5)(sin90o)

2. The index finger points in the direction of motion, the other fingers north to south in the direction of the magnetic field, leaving the thumb, showing the direction of the force pointing into the page, or “downwards” or negative z.

Bellwork 04/06/10Determine the centripetal force acting upon a 40.0 -kg child who makes 10 revolutions around the Cliffhanger ride at the amusement park in 29.3 seconds. The radius of the barrel is 2.90 meters.

Answer: Fnet = 533 NGiven: m = 40 kg; R = 2.90 m; T = 2.93 s (since 10 cycles takes 29.3 s).First, find speed usingspeed=(2 • pi • R) / T = 6.22 m/s. Then find the acceleration usinga = v2 / R = = (6.22 m/s)2 / (2.90 m) = 13.3 m/s/s Now use Fnet = m • a to find that Fnet = 533 N.

Simple Harmonic Motion• SHM: The motion of an

oscillating object which experiences a displacement and a restoring force

• Displacement, x: the distance of an object from its equilibrium position

SHM Descriptions• Amplitude, A: the magnitude of the

maximum displacement 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.

Energy and Speed in SHM

• A wave is defined as the transfer of energy from one point to another

• Total Energy = Potential Energy + Kinetic Energy• Mechanical waves require a medium for the

transfer of energy to occur• Non-mechanical waves do not require a medium

for the transfer of energy to occur

Calculating Energy in SHM

• Etotal = KE + PE or E = K + U• Consider the case of a block of mass m attached

to a massless spring. As the block oscillates (without friction), the spring provides the restorative force that moves it from + x to – x.

Simple Harmonic Motion of a Block on a Spring

(see [. 435)

Calculating the Spring Force• Such springs are said to follow Hooke's Law.

Fspring = k * x

where k is the spring constant and x is the amount of stretch or compression.

• If a spring is not stretched or compressed, then there is no elastic potential energy stored in it.

• The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it.

Calculating Elastic Potential Energy

• The amount of elastic potential energy is related to the amount of stretch (or compression) and the spring constant. The equation is

The work done by an external force in stretching or compressing a spring (to overcome the spring force) is calculated by:

Where k is the spring constant and x is the stretch or compression distance from the equilibrium position

2

2

1kxW

Calculating Energy in SHM

• KE = ½ mv2 and PE = ½ kx2

• So that at the maximum displacement, x= A, or the amplitude, it’s instantaneous velocity is 0.

• Etotal = ½ m(0)2 + ½k(A)2 = ½k(A)2

• The total energy of an object in SHM is directly proportional to the square of the amplitude

Calculating Velocity in SHM

• Velocity of the oscillating object is a function of the position

½k(A)2 = ½ mv2 + ½ kx 2 .• Solving for v allows us to find the velocity

anywhere along the path

Example 1: A hummingbird beats its wings up and down with a frequency of 80.0 Hz. What is the period of the hummingbird’s flaps?

Example 2: In anticipation of her first game, Alesia pulls back the handle of a pinball machine a distance of 5.0 cm. The force constant of the spring in the handle is 200 N/m. How much force must Alesia exert?

Example 3: A block with a mass of 0.25 kg sitting on a frictionless surface is connected to a spring that has a spring constant of 180 N/m. If the block is displaced 15 cm from its equilibrium position and released, what is a) the total energy of the system? And b) the speed of the block when it is at + 10cm?

• Please sketch the problem, identify the given information, and write the formula you choose to solve each problem.

Example 4 – Pendulums: A tall thin tree sways back and forth in the breeze with a frequency of 2 Hz. What is the period of the tree?

Example 5: Word-reknowned hypnotist Paulbar the Great swings his watch from a 20.0 cm chain in front of a subjects eyes. What is the period of swing of the watch?

Example 6: A spider swings in the breeze from a silk thread with a period of 0.6 s. How long is the spider’s strand of silk?

1. The wavelength of the wave in the diagram above is given by letter ______.

2. The amplitude of the wave in the diagram

above is given by letter _____. 3. Indicate the interval which represents one full

wavelength.

SHM in Circular Motion

An object in orbit also travels from

+ A to –A

Equations of Motion• SHM is described by a sinusoidal function of time

where ω is the angular speed of the object (in rad/s), or angular frequency, since ω = 2f

y = A sinθ(2f)t = A sin θ(2/T)t• The period of motion is given by

• or for a pendulum

where k is the spring force constant

• The frequency and the angular velocity are then

Speed and Acceleration

• The velocity of an object in simple harmonic motion is:

• and the acceleration:

• Note that the acceleration is not constant—the equations of motion for constant acceleration cannot be used here.

θ

θ

Wave Motion – a wave transfers energy from a source

• Amplitude, A : the height of a wave crest or trough• Wavelength, : the distance between two

successive troughs (or crests)• Frequency, f: the number of cycles, or

wavelengths, that pass by a given point per second• Period, T: The time for one complete wavelength

to pass by a given point• Wave speed, v (nu): The distance a wave travels in

a time of one period, T

Types of Waves

Mechanical: Travel through a medium (like sound waves or earthquakes)

Nonmechanical: Do not require a medium (like electromagnetic waves)

• Transverse• Longitudinal• Elliptical• Torsional

Transverse Waves – waves in which the particles vibrate at right angles to the direction of the wave's velocity

• Types of waves• http://dev.physicslab.org/DocumentPrint.aspx

?doctype=3&filename=WavesSound_IntroductionWaves.xml Tacoma Narrows Bridge Collapse

• http://video.google.com/videoplay?docid=4087615334344625698#docid=2260354895340450887