A 24.0 kg mass is attached to the bottom of a vertical spring, causing it to stretch 15.0 cm. a)...

A 24.0 kg mass is attached to the bottom of a vertical spring, causing it to stretch 15.0 cm.

a) What is the spring constant?b) What is the final potential energy stored in

the spring?

Warm-Up: January 26, 2015

75-100: A 65-75: B 55-65: C 45-55: D 0-45: F

First Semester Grading Scale

You must wear closed-toe shoes.

Lab Tomorrow

Oscillatory Motion and WavesOpenStax Chapter 16

When an elastic object is deformed, it experiences a restoring force.◦ Elastic means that it is capable of returning to its

original shape/size.◦ Includes springs, plastic rulers, rubber bands,

guitar strings, etc. The restoring force is given by Hooke’s Law

Hooke’s Law Revisited

xkF

The force constant, k, is related to the rigidity (or stiffness) of a system.

Larger k greater restoring force stiffer system

Measured in N/m The slope of a Force vs. Displacement graph

(if the graph is linear)

Force Constant

What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0 kg person gets in?

You-Try 16.1

What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0 kg person gets in?

You-Try 16.1

6.53x104 N/m

Work must be done in any deformation. Assuming no energy is lost to heat, sound,

etc., then all work is transferred to potential energy.

Since the force increases linearly with displacement, we can easily calculate the potential energy.

Energy in Hooke’s Law

Work Potential Energy

a) How much energy is stored in the spring of a tranquilizer gun that has a force constant of 50.0 N/m and is compressed 0.150 m?

b) If you neglect friction and the mass of the spring, at what speed will a 2.00 g projectile be ejected from the gun?

You-Try 16.2

a) 0.563 Jb) 23.7 m/s

Read Section 16.1 of OpenStax textbook (online): pages 555-559

Assignment

A force of 20.0 N is applied to the tip of a ruler, causing it to deflect 4.00 cm. What is the force constant of the ruler?


You will be assigned to a group. Your group’s goal is to determine the

relationship between the following:◦ The mass at the end of the string◦ The length of the string◦ The period of an oscillation (the time it takes to

complete one oscillation).◦ The angle of oscillation

Materials allowed:◦ String (and scissors to cut the string)◦ Masses◦ Rulers/Metersticks/Protractor◦ Stopwatch (cell phone)

Be sure to record everything in your lab notebook!

Today’s Lab

Group 1 Uriostegui Parra, Karla Peek, Angela Sutton, Foster

Group 2Hy, Kevin

Galac, Roger Regge Jezycki, Jocelyn

Group 3 Moreno, Mark To, Frank Cdebaca, PaulGroup 4 Gorman,

CourtneyNguyen, Christina Dolphin, Jeremy

Group 5 Lee, Justin Nguyen, Peter Lenhoff, ShaneGroup 6

Kibret, Elroi Basinger, ShelbyNava Saucedo, Hector

Group 7Le, Tiffany

Garcia-Ayala, Diana Kirk, John

Group 8 Randazzo, Joseph Wu, Tong Lenhoff, Erin

Group 9 Lopez, Isabel Nguyen, Phat

Lab Groups

Group member names Descriptive lab name Materials used Procedures followed Data obtained Calculations made Conclusions reached Summary

Lab Reports Must Include

A 500.0 g mass is attached to a rubber band, causing it to stretch 5.00 cm. How much potential energy is stored in the rubber band?


Period and Frequency in Oscillations

OpenStax section 16.2AP Physics 1

Periodic motion is motion that repeats itself at regular time intervals.

Examples:◦ Vibrating guitar string◦ Pendulum◦ Mass bobbing at the end of a spring

Periodic Motion

The period, T, is the time to complete one oscillation.◦ Measured in seconds

The frequency, f, is the number of oscillations per unit time.◦ Measured in Hertz

Period and Frequency

s

1

second

cycle 1Hz 1

Tf

1

A medical imaging device produces ultrasound by oscillating with a period of 0.400 μs. What is the frequency of this oscillation?

Example 16.3

2.50x106 Hz

The frequency of middle C on a typical musical instrument is 264 Hz. What is the time for one complete oscillation?

You-Try 16.3

3.79 ms

Identify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event.

Think-Pair-Share

You have the remainder of class to work on your lab report with your lab group.

Lab

A stroboscope is set to flash every 8.00x10-5 s. What is the frequency of the flashes?


Simple Harmonic Motion:

A Special Periodic Motion


Simple harmonic motion is oscillatory motion for a system where the net force can be described by Hooke’s Law.◦ Pendulums are sometimes considered simple

harmonic motion – only for small angles. Equal/symmetric displacement on either

side of the equilibrium position. The maximum displacement from

equilibrium is called the amplitude.

Simple Harmonic Motion

Simple Harmonic Motion Example

Period and frequency are independent of amplitude, so simple harmonic oscillators can be used as clocks.

They are a good analogue for waves, including invisible ones (sound, electromagnetic)

Importance of SHM

For simple harmonic oscillators:

What do period and frequency not depend on?

Period and Frequency

m

k

Tf

k

mT

2

11

2

If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping.

Calculate the frequency and period of these oscillations for such a car if the car’s mass (including its load) is 900. kg and the force constant of the suspension system is 6.53x104 N/m

You-Try 16.4

f=1.36 HzT=0.738 s

Simple Harmonic Motion and Waves

Simple Harmonic Motion and Waves

T

tXtx

2cos

m

kX

T

Xv

T

tv

T

t

T

Xtv

2

2sin

2sin

2

max

max

txm

k

T

t

m

kXta

2

cos

Example

Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the period, frequency, and amplitude of the sound waves as the volume decreases.

Think-Pair-Share

A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic motion is 1.05 s. What is his mass if the mass of the board is negligible?

You-Try #18

Read OpenStax Chapter 16 (through 16.3) OpenStax page 588 #1-7 OpenStax page 590 #1-21 odd

Assignments

If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?


One person from each group collect lab notebooks from all group members and turn them in.

Lab Reports Due

Homework Questions?

The Simple PendulumOpenStax section 16.4

AP Physics 1

Small-diameter mass, called the pendulum bob

String has negligible mass, but strong enough to not stretch appreciably

Undergoes simple harmonic motion if θ<15°

The Simple Pendulum

Simple PendulumsinmgF

mgF

L

smgF

sL

mgF

kxF

L

mgk

k

mT 2

Lmg

mT

/2

Period of Simple Pendulum

g

LT 2

Does not depend on mass. Does not depend on amplitude (for θ<15°) Can be finely adjusted, and can make

excellent clocks. Can also be used to solve for g.

Period of Simple Pendulum

What is the acceleration due to gravity in a region where a simple pendulum having a length of 75.000 cm has a period of 1.7357 s?

You-Try 16.5

What is the effect on the period of a pendulum if you double its length?

Think-Pair-Share

What is the length of a pendulum that has a period of 0.500 s? Let g=9.80 m/s2.

You-Try 22

The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency? Let g=9.80 m/s2.

You-Try 26


Assignments

Punxsutawney Phil, seer of seers, prognosticator of prognosticators, has an extremely accurate pendulum clock in his secret lair. The period of this clock is exactly six weeks.

a) What is the length of this pendulum clock?b) Is your answer realistic? Why or why not?

Warm-Up: February 2, 2015

New date and time Wednesday, February 25 Time TBD

American Mathematics Competition

Homework Questions?

Energy and the Simple Harmonic

OscillatorOpenStax Section 16.5

AP Physics 1

Energy is conserved

Maximum speed occurs at equilibrium position.

Simple Harmonic Oscillators

constant2

1

2

1

constant

22

kxmv

UK

Xm

kv max

Energy is conserved

Maximum speed occurs at equilibrium position.

Pendulums (small θ)

constant2

1

2

1

constant

222

mgLmL

UK

maxmax L

gv

Suppose that a car is 900. kg and has a suspension system that has a force constant of 65.3 kN/m. The car hits a bump and bounces with an amplitude of 0.100 m. What is its maximum velocity (assuming no damping)?

You-Try 16.6

0.852 m/s

Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.00x105 kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven; the driving force is transferred to the object, which oscillates instead of the entire building.

a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s?

b) What energy is stored in the springs for a 2.00 m displacement from equilibrium?

You-Try 36

Uniform Circular Motion and Simple Harmonic

MotionOpenStax section 16.6

AP Physics 1

An object moving in uniform circular motion can be used as an analogue for simple harmonic motion.

Circular Motion Simple Harmonic Motion

sin,cos, yx

The Equations

k

mT

X

xvv

T

tXtx

2

1

2cos

2

2

max

A ladybug sits 12.0 cm from the center of a Beatles album spinning at 33.3 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting sun?

You-Try 40

0.419 m/s


Assignments

a) If the spring stretches 0.250 m while supporting an 8.00 kg child, what is its spring constant?

b) What is the time for one complete bounce?c) What is the child’s maximum velocity if the

amplitude of her bounce is 0.200 m?


The device pictured entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring.

Homework Questions?

Damped Harmonic Motion

OpenStax Section 16.7AP Physics 1

Friction is not always negligible. Damping is the slowing and stopping of

oscillations, caused by a non-conservative force (such as friction).

Damping is sometimes part of a design (such as a car’s shock absorbers).

For small damping, the amplitude slowly decreases while period and frequency are nearly unchanged.

Friction is Real!

Non-conservative work removes mechanical energy (usually to thermal energy).

Damping

UKW nc

Large damping causes the period to increase and the frequency to decrease.

Very large damping prevents oscillation – the system just returns to equilibrium.

Critical damping is the amount of damping that returns a system to equilibrium as quickly as possible.

Overdamped systems return to equilibrium slower than critical damping.

Large Damping

Which is critical damping? Which is overdamping?

Think-Pair-Share

Car shock absorbers Bathroom scale

Applications of Critical Damping

Suppose a 0.200 kg object is connected to a spring as shown, but there is simple friction between the object and the surface, and the coefficient of kinetic friction is equal to 0.0800. The force constant of the spring is 50.0 N/m. Use g=9.80 m/s2.

a) What is the frictional force between the surfaces?

b) What total distance does the object travel if it is released from rest 0.100 m from equilibrium?

You-Try 16.7

A novelty clock has a 0.0100 kg mass object bouncing on a spring that has a force constant of 1.25 N/m.

a) What is the maximum velocity of the object if the object bounces 3.00 cm above and below the equilibrium position?

b) How many Joules of kinetic energy does the object have at its maximum velocity?


Homework Questions?

Forced Oscillations and Resonance


What do you have to do to swing high on a swing?

Think-Pair-Share

The natural frequency is the frequency at which it would oscillate if there were no driving and no damping force.

If you drive a system at a frequency equal to its natural frequency, its amplitude will increase. This is called resonance.◦ A system being driven at its natural frequency is

said to resonate.

Natural Frequency

The highest amplitude oscillations occur when:◦ The system is driven at its natural frequency◦ There is minimal damping

Resonance

A famous trick involves a performer singing a note toward a crystal glass until the glass shatters. Explain why the trick works in terms of resonance and natural frequency.

Think-Pair-Share

Tacoma Narrows Bridge

A suspension bridge oscillates with an effective force constant of 1.00x108 N/m.

a) How much energy is needed to make it oscillate with an amplitude of 0.100 m?

b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart 1.00x104 J of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude?

You-Try 46


Assignments

How much energy must the shock absorbers of a 1200 kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.


Homework Questions?

Tentatively scheduled for April 18

Mock AP Test

Wednesday, February 25 4th period and lunch

AMC

WavesOpenStax section 16.9

AP Physics 1

A wave is a disturbance that propagates, or moves from the place it was created.

Waves carry energy, not matter. Similar to simple harmonic motion, waves

have a period, frequency, and amplitude. Waves also have a wave velocity, the

velocity at which the disturbance moves. Waves also have a wavelength, λ, the

distance between identical parts of the wave.

Waves

Example Wave

Wave Speed

fvT

v

w

w

Calculate the wave velocity of an ocean wave if the distance between wave crests is 10.0 m and the time for a sea gull to bob up and down is 5.00 s.

You-Try 16.8

2.00 m/s

Transverse vs. Longitudinal

In transverse waves, also called shear waves, the direction of energy transfer and the direction of displacement are perpendicular.◦ Examples: strings on musical instruments, light

In longitudinal waves, also called compressional waves, the direction of energy transfer and the direction of displacement are parallel.◦ Example: sound

Some waves, such as ocean waves, are a combination of transverse and longitudinal.

Transverse vs. Longitudinal

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at 0.800 m/s?

You-Try 52

Superposition and InterferenceOpenStax section 16.10

AP Physics 1

Most real-world waves are combinations of simple waves.

When two or more waves arrive at the same point, their disturbances are added together. This is called superposition.

In constructive interference, crest meets crest, and trough meets trough, and the resultant is a wave with a larger amplitude.

In destructive interference, crest meets trough, and resulting amplitude is smaller than either original amplitude.◦ Amplitude is zero for pure destructive interference.

Superposition

Pure Constructive Interference

Pure Destructive Interference

wave-on-a-string_en.jar

PhET Waves on a String

Radio waves transmitted through space at 3.00x108 m/s by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?


More Superposition

Sometimes waves superimpose in a way that causes an apparent lack of sideways motion.

These waves are called standing waves. Standing waves have points that do not

move, called nodes. The points that move the most are called

antinodes.

Standing Waves

Green dots are nodes

PhET Wave on a String

The fundamental frequency is the lowest possible frequency for a standing wave.

Overtones or harmonics are multiples of the fundamental frequency.

Fundamental Frequency

Overtones

The superposition of two waves of similar frequency has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats, in the beat frequency.

Beats

21 fffB

The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?

You-Try 58

Twin jet engines on an airplane are producing an average sound frequency of 4100.000 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?

You-Try 60

Energy in Waves: Intensity


Wave energy is related to wave amplitude

The intensity, I, of a wave is the power, P, carried through area A.

Wave Energy

2kxFxW

kxF

A

PI

Valid for any flow of energy. Has units of W/m2. Other intensity units include decibels.

◦ 90 decibel = 10-3 W/m2

Intensity

The average intensity of sunlight on Earth’s surface is about 700. W/m2.

a) Calculate the amount of energy that falls on a solar collector having an area of 0.500 m2 in 4.00 hours.

b) What intensity would such sunlight have if concentrated by a magnifying glass onto an area 200. times smaller than its own?

You-Try 16.9

If two identical waves, each having an intensity of 1.00 W/m2, interfere perfectly constructively, what is the intensity of the resulting wave?


Read OpenStax Chapter 16 OpenStax page 588 #1-18 OpenStax page 590 #1-71 odd

Assignments

Homework Questions?

With your partner, write 3 multiple choice and/or multiple select questions. Indicate the correct answer(s). For computational questions, show work leading to the correct answer(s).

Test Questions

A 24.0 kg mass is attached to the bottom of a vertical spring, causing it to stretch 15.0 cm. a)...

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