Waves, Optics, Waves, Optics, Oscillation, and Oscillation, and

GravitationGravitation

By: Charnae’ Kearney By: Charnae’ Kearney

and and Andy Hurst Andy Hurst

Traveling WaveTraveling Wave

Any kind of wave which propagates in a Any kind of wave which propagates in a single direction with negligible change in single direction with negligible change in shape.shape.

Traveling waves are observed when a Traveling waves are observed when a wave is not confined to a given space wave is not confined to a given space along the medium. The most commonly along the medium. The most commonly observed traveling wave is an ocean wave observed traveling wave is an ocean wave

Traveling and Standing WavesTraveling and Standing Waves An important class of traveling waves is plane waves in air which create An important class of traveling waves is plane waves in air which create

standing waves in rectangular enclosures such as ``shoebox'' shaped standing waves in rectangular enclosures such as ``shoebox'' shaped concert halls. concert halls.

Standing waves don't go anywhere, but they do have regions where the Standing waves don't go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are disturbance of the wave is quite small, almost zero. These locations are called nodes. There are also regions where the disturbance is quite intense, called nodes. There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called antinodes. greater than anywhere else in the medium, called antinodes.

Wave PropagationWave Propagation Any of the waves that waves travelAny of the waves that waves travel With respect to the direction of the oscillation relative to the propagation With respect to the direction of the oscillation relative to the propagation

direction, we can distinguish between longitudinal wave and transverse direction, we can distinguish between longitudinal wave and transverse waves.waves.

For electromagnetic waves, propagation may occur in a vacuum as well as For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Most other wave types cannot propagate through in a material medium. Most other wave types cannot propagate through vacuum and need a transmission medium to exist.vacuum and need a transmission medium to exist.

Another useful parameter for describing the propagation is the wave Another useful parameter for describing the propagation is the wave velocity that mostly depends on some kind of density of the medium.velocity that mostly depends on some kind of density of the medium.

Principle of SuperpositionPrinciple of Superposition

The regions where they overlap, the The regions where they overlap, the resultant displacement is the algebraic resultant displacement is the algebraic sum of their separate displacements.sum of their separate displacements.

Simple Harmonic MotionSimple Harmonic Motion Regular, repeated, friction-free motion in which Regular, repeated, friction-free motion in which

the restoring force has the mathematical form the restoring force has the mathematical form F= -kxF= -kx

Common examples: mass on a spring and a Common examples: mass on a spring and a pendulumpendulum

The word “harmonic” refers to the motion being The word “harmonic” refers to the motion being sinusoidal, it is “simple” when there is pure sinusoidal, it is “simple” when there is pure sinusoidal motion of a single frequencysinusoidal motion of a single frequency

As an object vibrates in harmonic motion, energy As an object vibrates in harmonic motion, energy is transferred between potential and kinetic is transferred between potential and kinetic energy.energy.

Mass on a SpringMass on a Spring

When it vibrates it has both a period and a When it vibrates it has both a period and a frequencyfrequency

Restoring force – the force trying to Restoring force – the force trying to restore it (mass on a spring) back towards restore it (mass on a spring) back towards the center of the oscillationthe center of the oscillation

PendulumPendulum

A mass on the end of a string which A mass on the end of a string which oscillates in harmonic motionoscillates in harmonic motion

T= 2T= 2π√π√L/GL/G L is the length of the pendulum L is the length of the pendulum G is the acceleration due to gravityG is the acceleration due to gravity

Newton’s Law of GravityNewton’s Law of Gravity

Every point mass in the universe attracts Every point mass in the universe attracts every other point mass with a force that is every other point mass with a force that is directly proportional to the product of their directly proportional to the product of their masses and inversely proportional to the masses and inversely proportional to the square of the distance between them.square of the distance between them.

F= G mF= G m11 m m22

--------------------------------------------------

rr²²

                ,

F is the force between the masses, G is the gravitational constant,

m1 is the first mass, m2 is the second mass, and

r is the distance between the masses.

Newton’s Law of Gravity Contin.Newton’s Law of Gravity Contin.

Gravitation is a Gravitation is a UNIVERSALUNIVERSAL force force between all objects in the universe.between all objects in the universe.

Circular Orbits of Planets & SatellitesCircular Orbits of Planets & Satellites As a satellite orbits the earth, it is pulled toward As a satellite orbits the earth, it is pulled toward

the earth with a gravitational force which is the earth with a gravitational force which is acting as a centripetal force. The inertia of the acting as a centripetal force. The inertia of the satellite causes it to tend to follow a straight-line satellite causes it to tend to follow a straight-line path, but the centripetal gravitational force pulls path, but the centripetal gravitational force pulls it toward the center of the orbit. it toward the center of the orbit.

If a satellite of mass If a satellite of mass mm moves in a circular orbit moves in a circular orbit around a planet of mass around a planet of mass MM, we can set the , we can set the centripetal force equal to the gravitational force centripetal force equal to the gravitational force and solve for the speed of the satellite orbiting at and solve for the speed of the satellite orbiting at a particular distance a particular distance rr::

r

GMv

r

GmM

r

mv

FF Gc

2

2

General Orbits of Planets & General Orbits of Planets & SatellitesSatellites

Elliptical Motion:Elliptical Motion:

Kepler’s Law of Planetary MotionKepler’s Law of Planetary Motion The orbit of every planet is an ellipse with the The orbit of every planet is an ellipse with the

Sun at one of the two foci. Sun at one of the two foci. A line joining a planet and the Sun sweeps out A line joining a planet and the Sun sweeps out

equal areas during equal intervals of time.equal areas during equal intervals of time.[1] The square of the orbital period of a planet is The square of the orbital period of a planet is

directly proportional to the cube of the semi-directly proportional to the cube of the semi-major axis of its orbit major axis of its orbit

Important Key Terms to RememberImportant Key Terms to Remember

PeriodPeriod- (T) of the motion is the time required for the - (T) of the motion is the time required for the motion to repeat.motion to repeat.

FrequencyFrequency- (f) refers to the number of complete - (f) refers to the number of complete repetitions of the motion that occur each second. The repetitions of the motion that occur each second. The frequency is inversely related to the period.frequency is inversely related to the period.

Simple harmonic motionSimple harmonic motion- (SHM) refers to periodic - (SHM) refers to periodic vibrations or oscillations that exhibit two characteristics: vibrations or oscillations that exhibit two characteristics: 1) the force acting on the object and the magnitude of 1) the force acting on the object and the magnitude of the object’s acceleration are always directly proportional the object’s acceleration are always directly proportional to the displacement of the object from its equilibrium to the displacement of the object from its equilibrium position, and 2) both the force vector and the position, and 2) both the force vector and the acceleration vector are directed opposite to the acceleration vector are directed opposite to the displacement vector and therefore in toward the object’s displacement vector and therefore in toward the object’s equilibrium position.equilibrium position.

Important Key Terms to Remember Important Key Terms to Remember Cont.Cont.

Simple pendulumSimple pendulum- is assumed to have its entire mass - is assumed to have its entire mass concentrated at the end of its length. The simple concentrated at the end of its length. The simple pendulum undergoes SHM if the maximum angle that it pendulum undergoes SHM if the maximum angle that it is displaced from equilibrium is small (approximately 15is displaced from equilibrium is small (approximately 15°° or less).or less).

Principle of superpositionPrinciple of superposition- states that when two waves - states that when two waves pass through a medium at the same time, the resultant pass through a medium at the same time, the resultant displacement of the medium at any particular moment of displacement of the medium at any particular moment of time equals the algebraic sum of the displacement of the time equals the algebraic sum of the displacement of the component waves at that point.component waves at that point.

Standing wavesStanding waves- are produced by the superposition of - are produced by the superposition of two periodic waves having identical frequencies and two periodic waves having identical frequencies and amplitudes which are traveling in opposite directions.amplitudes which are traveling in opposite directions.

Important Formulas

K=(F/x) f=(1/T) V=f F=-kx T=2π(m/k) T=2π(L/g) V=λf (wave velocity is = to the product of

wavelength and frequency)

Example #1Example #1 A spring of constant A spring of constant kk = 100 N/m hangs at its natural length from a fixed stand. A = 100 N/m hangs at its natural length from a fixed stand. A

mass of 3 kg is hung on the end of the spring, and slowly let down until the spring and mass of 3 kg is hung on the end of the spring, and slowly let down until the spring and mass hang at their new equilibrium position. mass hang at their new equilibrium position. xx

(a) Find the value of the quantity (a) Find the value of the quantity xx in the figure above. in the figure above.

The spring is now pulled down an additional distance The spring is now pulled down an additional distance xx and released from rest. and released from rest. (b) What is the potential energy in the spring at this distance?(b) What is the potential energy in the spring at this distance? (c) What is the speed of the mass as it passes the equilibrium position?(c) What is the speed of the mass as it passes the equilibrium position? (d) How high above the point of release will the mass rise?(d) How high above the point of release will the mass rise? (e) What is the period of oscillation for the mass?(e) What is the period of oscillation for the mass?

xx

Example #1 SolutionExample #1 Solution (a) As it hangs in equilibrium, the upward spring force must be equal and opposite to the (a) As it hangs in equilibrium, the upward spring force must be equal and opposite to the

downward weight of the block.downward weight of the block.

(b) The potential energy in the spring is related to the displacement from equilibrium position (b) The potential energy in the spring is related to the displacement from equilibrium position by the equationby the equation

(c) Since energy is conserved during the oscillation of the mass, the kinetic energy of the (c) Since energy is conserved during the oscillation of the mass, the kinetic energy of the mass as it passes through the equilibrium position is equal to the potential energy at the mass as it passes through the equilibrium position is equal to the potential energy at the amplitude. Thus,amplitude. Thus,

(d) Since the amplitude of the oscillation is 0.3 m, it will rise to 0.3 m above the equilibrium (d) Since the amplitude of the oscillation is 0.3 m, it will rise to 0.3 m above the equilibrium position. position.

(e) (e)

m

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Example #2Example #2 A string is attached to a vibrating machine which has a frequency of 120 Hz. The other A string is attached to a vibrating machine which has a frequency of 120 Hz. The other

end of the string is passed over a pulley of negligible mass and friction and is attached to end of the string is passed over a pulley of negligible mass and friction and is attached to a weight hanger which holds a mass a weight hanger which holds a mass mm = 0.5 kg. = 0.5 kg.

(a) Determine the tension in the string. (a) Determine the tension in the string. (b) The speed of the wave in the string is related to the tension by the equation (b) The speed of the wave in the string is related to the tension by the equation , where , where FFTT is the tension in the string and is the tension in the string and μμ is the linear density of the string. If the linear density of is the linear density of the string. If the linear density of

this string is 0.05 kg/m, determine the speed of the wave in the string.this string is 0.05 kg/m, determine the speed of the wave in the string. (c) Determine the wavelength of the wave in the string.(c) Determine the wavelength of the wave in the string. (d) Determine the length of the string from the point of attachment on the vibrating machine to the (d) Determine the length of the string from the point of attachment on the vibrating machine to the

pulley.pulley. (e) Would you need to increase or decrease the mass on the hanger to produce a lower number of (e) Would you need to increase or decrease the mass on the hanger to produce a lower number of

loops? Explain.loops? Explain.

m

L

m

L

TFv

Example #2 SolutionExample #2 Solution

(a) (a) (b)(b)

(c) (c) (d) (d) (e) A lower number of loops would imply a longer (e) A lower number of loops would imply a longer

wavelength, which would require a higher speed, wavelength, which would require a higher speed, which would require a higher tension in the string, which would require a higher tension in the string, which would require increasing the mass on the which would require increasing the mass on the hanger. hanger.

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5

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Example #3Example #3 A pendulum of mass 0.4 kg and length 0.6 m is pulled back and A pendulum of mass 0.4 kg and length 0.6 m is pulled back and

released from and angle of 10˚ to the vertical. released from and angle of 10˚ to the vertical.

(a) What is the potential energy of the mass at the instant it is (a) What is the potential energy of the mass at the instant it is released. Choose potential energy to be zero at the bottom of the released. Choose potential energy to be zero at the bottom of the swing. swing.

(b) What is the speed of the mass as it passes its lowest point?(b) What is the speed of the mass as it passes its lowest point?

   This same pendulum is taken to another planet where its period is This same pendulum is taken to another planet where its period is

1.0 second. 1.0 second. (c) What is the acceleration due to gravity on this planet?(c) What is the acceleration due to gravity on this planet?

Example #3 SolutionExample #3 Solution (a) First we must find the height above the lowest point in the swing at the (a) First we must find the height above the lowest point in the swing at the

instant the pendulum is released.instant the pendulum is released.

    Recall from chapter 1 of this study guideRecall from chapter 1 of this study guide

that that

Then Then

   (b) Conservation of energy:(b) Conservation of energy:

(c) (c)

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4.022

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