Physics - Chapter 12

Vibrations & Waves

Lesson 62: Hooke’s Law and Simple Harmonic Motion

Notes

NOTES

  • The force provided by the spring is governed by Hooke’s Law                        {F_s} =  - kx
  • k is spring constant, or sometimes called a force constant.
  • The units are Newtons per meter.
  • x will be negative when the spring is stretched
  • x will be positive when the spring is compressed
  • (-) because the spring is always providing a restoring force opposite the motion of the mass
  • The (-) belongs to the x, not the k.

  • Simple harmonic motion (SHM) occurs when the net force acting in the direction of motion follows Hooke’s Law.
  • SHM will repeat a cycle of back and forth along same path forever.
  • Also called periodic motion.

  • Remember elastic potential energy can be found                                        P{E_{elastic}} = \frac{1}{2}k{x^2}
  • The energy is only stored in a spring when it is either stretched or compressed.
  • The potential energy in a spring is always positive.

http://www.kshitij-iitjee.com/Study/Physics/Part1/Chapter13/42.jpg

EXAMPLE 1 – Solve the following using Hooke’s Law.

  1. A pinball machine uses a spring that is compressed 4.0 cm to launch a ball. If the spring constant is 13 N/m, what is the force on the ball at the moment the spring is released?
  1. A load of 45 N attached to a spring that is hanging vertically stretches the spring 0.14 m. What is the spring constant?

EXAMPLE 2 - A mass-spring system oscillates with an amplitude of 3.5 cm. The spring constant is 250 N/m and the
mass is 0.50
kg.

  1. Find the force on the mass at the instant the spring is released.
  1. Calculate the maximum acceleration of the mass-spring system.
  1. Find the maximum potential energy of the mass-spring system.

EXAMPLE 2 –Use the figure below to fill in the missing energy quantities.

http://www.physicsclassroom.com/Class/waves/u10l0dq4.gif

Physics - Chapter 12

Lesson 63: Wave Types

Notes

NOTES 

  • A transverse wave is a wave that its particles move perpendicular to the overall motion of the wave. http://images.tutorvista.com/cms/images/95/wave.jpg
  • A longitudinal wave is a wave in which its particles move in the same direction as the overall motion of the wave.

  • Amplitude is the maximum distance object travels away from rest point
  • The variable for amplitude is a capital A
  • The units for amplitude are meters
  • Period is the time it takes to complete one full cycle of motion
  • The variable for period is a capital T
  • The units for the measuring the period are in secondshttp://www.ck12.org/flx/show/THUMB_POSTCARD/image/user%3Ack12editor/201207161342465367821350_59182fc3461be6fe8da30bb49acf83bd-201207161342466115774895.png
  • Frequency is the number of cycles per unit of time
  • Said another way as the number of waves past a given a point in one second
  • The variable for frequency is the lower case Latin letter                
  • The SI units for frequency is hertz (Hz)

f = \frac{1}{T}

  • Frequency is the inverse of the period                                

  • The wavelength of a wave is the distance between two successive points on the wave.
  • Typically measured from crest-to-crest.
  • The variable used to represent wavelength is the lowercase Greek letter lambda, λ.
  • Wavelength should be measured in meters.

EXAMPLE 1 – Use the figure at right to answer the following. http://www.ethiopianteachers.org/EATOnlineResource/Subjects/Physics/EBook/Image/UnitFour_clip_image028.jpg

  1. Find the amplitude of the wave.
  1. Measure the wavelength.
  1. If the hand that generated the wave moved up and down two times in 8.0 s, what is the period of the wave?
  1. What is the frequency of the wave?
  1. What is the speed of the wave?

HOMEWORK -



Physics - Chapter 12

Lesson 64: Pendulums

Notes

NOTES

  • A pendulum also exhibits simple harmonic motion under certain conditions.http://img.sparknotes.com/content/testprep/bookimgs/sat2/physics/0011/pendulum_FBD.gif
  • The restoring force to maintain simple harmonic motion acts tangential to the path of the swing.

  • A pendulum exhibiting simple harmonic motion will have a period that depends on:
  • longer the pendulum, longer the arc (larger amplitude)
  • acceleration due to gravityhttp://img.sparknotes.com/content/testprep/bookimgs/sat2/physics/0011/pendulum_summary.gif
  • The faster gravity can pull the pendulum, the shorter the time it takes to complete its cycle (period)

T = 2\pi \sqrt {\frac{L}{g}}

  • To find the frequency of the wave, find the period and takes its inverse, or use

f = \frac{1}{T} = \frac{1}{{2\pi }}\sqrt {\frac{g}{L}}

  1. mass of the pendulum bob, or
  2. amplitude of the swing

EXAMPLE 1 – Use the figure at right to answer the following. https://www.assistments.org/images/assistments/16017.jpg

  1. At which location will the pendulum experience the greatest amount of gravitational potential energy?
  1. At which location will the pendulum experience the greatest amount of kinetic energy?
  1. At which location will the pendulum experience the greatest magnitude of force?
  1. At which location will the pendulum experience the greatest magnitude of acceleration?
  1. At which location will the pendulum experience the greatest magnitude of velocity?


EXAMPLE 3 – Solve the following.

  1. Calculate the period and frequency of a 3.500 m long pendulum swinging in Grant Park in Chicago, IL where the acceleration due to gravity is 9.803 .
  1. A trapeze artist swings in simple harmonic motion with a period of 3.8 s. Calculate the length of the cables supporting the trapeze artist.
  1. What is the free-fall acceleration in a location where the period of a 0.850 m long pendulum is 1.86 s?

HOMEWORK -


Physics - Chapter 12

Lesson 65: Mass-Spring System in Simple Harmonic Motion

Notes

NOTES

  • The period of a mass oscillating in simple harmonic motion depends on
  1. mass of the spring system, m
  • the force the spring can provide is set. So, F = ma tells us that acceleration will be less and kinematics says that time would be more!
  1. spring constant, k
  • stiffer the spring, the less time it takes to accelerate
  • It only depends on these two things because gravity remains constant and those are the only two values that could vary in the system.

T = 2\pi \sqrt {\frac{m}{k}}

  • Again, the frequency can be found by calculating the inverse of the period found in the equation above, or

f = \frac{1}{T} = \frac{1}{{2\pi }}\sqrt {\frac{k}{m}}

  • Notice, the period of a mass-spring system does not depend on:
  1. acceleration due to gravity, or
  2. amplitude of the spring

EXAMPLE 1 – A spring constant of 30.0 N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes.

  1. 2.3 kg
  1. 19 kg
  1. 15 g

HOMEWORK -


Physics - Chapter 12

Lesson 66: Wave Speed and Interference

Notes

LEARNING TARGET JOURNAL

  1. A wave pulse is sent down a rope that is attached to a pole in such a way that the end of the rope is free to slide relative to the pole.
  1. Identify where the end of the rope will be when the incident pulse reaches the pole.


  2. Draw the reflected wave.
  1. A wave pulse is sent down a rope that is fixed to the pole.
  1. Identify where the end of the rope will be when the incident pulse reaches the pole.

  2. Draw the reflected wave.

NOTES

  1. the velocity (v) is measured in meters per second because                        𝜆
  2. λ is the wavelength measured in meters, andhttp://media.wiley.com/Lux/48/313748.image1.jpg
  3. ƒ is the frequency measured in hertz
  • which is a per second measurement

  • If given the period of the wave, you must find the frequency as the inverse of the period before finding the speed of the wave.

  • Waves that are in phase have crests and valleys that line up exactly.
  • This type will make a wave of larger amplitude
  • Destructive interference occurs when two waves meet out of phase.
  • This will typically make a wave of smaller amplitude
  • This would mean that a crest of one wave would interfere with the trough of the other wave.
  • The result is that the waves will cancel each other out.

EXAMPLE 1 – Two wave pulses are sent towards each other in the same medium as illustrated in the following problems. Identify the type of interference and draw the resultant wave during interference.

Type of Interference:

                

Resultant Wave:

Type of Interference:

                

Resultant Wave:

  1. http://www.wired.com/images_blogs/wiredscience/2011/10/untitled_4.jpg

Type of Interference:

                

Resultant Wave:

  1. http://www.wired.com/images_blogs/wiredscience/2011/10/untitled_5.jpg

Type of Interference:

                

Resultant Wave:

EXAMPLE 2 – Which waveform of the three shown is the resultant wave? Explain how you identified the resultant wave.


        


        


        

EXAMPLE 3 – Solve the following.

  1. A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35 m. What value does this give for the speed of sound in air?
  1. You dip your finger into a pan of water twice each second, producing waves with crests that are separated by 0.15 m. Determine the speed of these surface waves.

HOMEWORK -

Physics - Chapter 12

Lesson X: Surface Waves & Pressure Waves

Notes

NOTES

  • A surface wave is a mechanical wave that travels along the surface, or boundary of two mediums.
  • Surface waves in fluid act as
  1. transverse waves near the top of the fluid, and
  2. longitudinal waves as you go deeper into the fluid.

  • A seismic wave is the type of wave associated with an earthquake.https://upload.wikimedia.org/wikipedia/commons/thumb/b/be/Speeds_of_seismic_waves.PNG/300px-Speeds_of_seismic_waves.PNG
  • Seismic waves actually come as two types
  1. P-waves, or primary waves, travel as longitudinal compression waves.
  • P-waves can travel the fastest, upwards of 14 km/s.
  1. S-waves, or secondary waves, follow p-waves and travel as transverse waves through loose portions of the Earth’s crust.

http://drjudywood.com/articles/DEW/dewpics/Image297.gif