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Chapter 15. Oscillatory Motion April 17 th , 2006. The last steps …. If you need to, file your taxes TODAY! Due at midnight. This week Monday & Wednesday – Oscillations Friday – Review problems from earlier in the semester Next Week Monday – Complete review. The FINAL EXAM.

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Chapter 15

Chapter 15

Oscillatory Motion

April 17th, 2006

Periodic Motion


The last steps
The last steps …

  • If you need to, file your taxes TODAY!

    • Due at midnight.

  • This week

    • Monday & Wednesday – Oscillations

    • Friday – Review problems from earlier in the semester

  • Next Week

    • Monday – Complete review.

Periodic Motion


The final exam
The FINAL EXAM

  • Will contain 8-10 problems. One will probably be a collection of multiple choice questions.

  • Problems will be similar to WebAssign problems but only some of the actual WebAssign problems will be on the exam.

  • You have 3 hours for the examination.

  • SCHEDULE: MONDAY, MAY 1 @ 10:00 AM

Periodic Motion


Things that bounce around

Things that Bounce Around

Periodic Motion


The simple pendulum
The Simple Pendulum

Periodic Motion


The spring
The Spring

Periodic Motion


Periodic motion
Periodic Motion

  • From our observations, the motion of these objects regularly repeats

    • The objects seem t0 return to a given position after a fixed time interval

  • A special kind of periodic motion occurs in mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position

    • If the force is always directed toward the equilibrium position, the motion is called simple harmonic motion

Periodic Motion


The spring for a moment
The Spring … for a moment

  • Let’s consider its motion at each point.

  • What is it doing?

    • Position

    • Velocity

    • Acceleration

Periodic Motion


Motion of a spring mass system
Motion of a Spring-Mass System

  • A block of mass m is attached to a spring, the block is free to move on a frictionless horizontal surface

  • When the spring is neither stretched nor compressed, the block is at the equilibrium position

    • x = 0

Periodic Motion


More about restoring force
More About Restoring Force

  • The block is displaced to the right of x = 0

    • The position is positive

  • The restoring force is directed to the left

Periodic Motion


More about restoring force 2
More About Restoring Force, 2

  • The block is at the equilibrium position

    • x = 0

  • The spring is neither stretched nor compressed

  • The force is 0

Periodic Motion


More about restoring force 3
More About Restoring Force, 3

  • The block is displaced to the left of x = 0

    • The position is negative

  • The restoring force is directed to the right

Periodic Motion


Acceleration cont
Acceleration, cont.

  • The acceleration is proportional to the displacement of the block

  • The direction of the acceleration is opposite the direction of the displacement from equilibrium

  • An object moves with simple harmonic motion whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium

Periodic Motion


Acceleration final
Acceleration, final

  • The acceleration is not constant

    • Therefore, the kinematic equations cannot be applied

    • If the block is released from some position x = A, then the initial acceleration is –kA/m

    • When the block passes through the equilibrium position, a = 0

    • The block continues to x = -A where its acceleration is +kA/m

Periodic Motion


Motion of the block
Motion of the Block

  • The block continues to oscillate between –A and +A

    • These are turning points of the motion

  • The force is conservative

  • In the absence of friction, the motion will continue forever

    • Real systems are generally subject to friction, so they do not actually oscillate forever

Periodic Motion


The motion
The Motion

Periodic Motion


Vertical spring
Vertical Spring

Equilibrium Point

Periodic Motion


Ye olde math
Ye Olde Math

Periodic Motion


Chapter 15

Periodic Motion


Example the spring
Example – the Spring from equilibrium (

Periodic Motion


Example the spring1
Example – the Spring from equilibrium (

Periodic Motion


Simple harmonic motion graphical representation
Simple Harmonic Motion – Graphical Representation from equilibrium (

  • A solution is x(t) = A cos (wt + f)

  • A, w, f are all constants

  • A cosine curve can be used to give physical significance to these constants

Periodic Motion


Simple harmonic motion definitions
Simple Harmonic Motion – Definitions from equilibrium (

  • A is the amplitude of the motion

    • This is the maximum position of the particle in either the positive or negative direction

  • w is called the angular frequency

    • Units are rad/s

  • f is the phase constant or the initial phase angle

Periodic Motion


Motion equations for simple harmonic motion
Motion Equations for Simple Harmonic Motion from equilibrium (

  • Remember, simple harmonic motion is not uniformly accelerated motion

Periodic Motion


Maximum values of v and a
Maximum Values of v and a from equilibrium (

  • Because the sine and cosine functions oscillate between ±1, we can easily find the maximum values of velocity and acceleration for an object in SHM

Periodic Motion


Graphs
Graphs from equilibrium (

  • The graphs show:

    • (a) displacement as a function of time

    • (b) velocity as a function of time

    • (c ) acceleration as a function of time

  • The velocity is 90o out of phase with the displacement and the acceleration is 180o out of phase with the displacement

Periodic Motion


Shm example 1
SHM Example 1 from equilibrium (

  • Initial conditions at t = 0 are

    • x (0)= A

    • v (0) = 0

  • This means f = 0

  • The acceleration reaches extremes of ± w2A

  • The velocity reaches extremes of ± wA

Periodic Motion


Shm example 2
SHM Example 2 from equilibrium (

  • Initial conditions at

    t = 0 are

    • x (0)=0

    • v (0) = vi

  • This means f = - p/2

  • The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A

Periodic Motion


Energy of the shm oscillator
Energy of the SHM Oscillator from equilibrium (

  • Assume a spring-mass system is moving on a frictionless surface

  • This tells us the total energy is constant

  • The kinetic energy can be found by

    • K = ½ mv 2 = ½ mw2A2 sin2 (wt + f)

  • The elastic potential energy can be found by

    • U = ½ kx 2 = ½ kA2 cos2 (wt + f)

  • The total energy is K + U = ½ kA 2

Periodic Motion


Energy of the shm oscillator cont
Energy of the SHM Oscillator, cont from equilibrium (

  • The total mechanical energy is constant

  • The total mechanical energy is proportional to the square of the amplitude

  • Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block

Periodic Motion


Energy of the shm oscillator cont1
Energy of the SHM Oscillator, cont from equilibrium (

  • As the motion continues, the exchange of energy also continues

  • Energy can be used to find the velocity

Periodic Motion


Energy in shm summary
Energy in SHM, summary from equilibrium (

Periodic Motion


Shm and circular motion
SHM and Circular Motion from equilibrium (

  • This is an overhead view of a device that shows the relationship between SHM and circular motion

  • As the ball rotates with constant angular velocity, its shadow moves back and forth in simple harmonic motion

Periodic Motion


Shm and circular motion 2
SHM and Circular Motion, 2 from equilibrium (

  • The circle is called a reference circle

  • Line OP makes an angle f with the x axis at t = 0

  • Take P at t = 0 as the reference position

Periodic Motion


Shm and circular motion 3
SHM and Circular Motion, 3 from equilibrium (

  • The particle moves along the circle with constant angular velocity w

  • OP makes an angle q with the x axis

  • At some time, the angle between OP and the x axis will be q = wt + f

Periodic Motion


Shm and circular motion 4
SHM and Circular Motion, 4 from equilibrium (

  • The points P and Q always have the same x coordinate

  • x (t) = A cos (wt + f)

  • This shows that point Q moves with simple harmonic motion along the x axis

  • Point Q moves between the limits ±A

Periodic Motion


Shm and circular motion 5
SHM and Circular Motion, 5 from equilibrium (

  • The x component of the velocity of P equals the velocity of Q

  • These velocities are

    • v = -wA sin (wt + f)

Periodic Motion


Shm and circular motion 6
SHM and Circular Motion, 6 from equilibrium (

  • The acceleration of point P on the reference circle is directed radially inward

  • P ’s acceleration is a = w2A

  • The x component is

    –w2A cos (wt + f)

  • This is also the acceleration of point Q along the x axis

Periodic Motion


Shm and circular motion summary
SHM and Circular Motion, Summary from equilibrium (

  • Simple Harmonic Motion along a straight line can be represented by the projection of uniform circular motion along the diameter of a reference circle

  • Uniform circular motion can be considered a combination of two simple harmonic motions

    • One along the x-axis

    • The other along the y-axis

    • The two differ in phase by 90o

Periodic Motion


Simple pendulum summary
Simple Pendulum, Summary from equilibrium (

  • The period and frequency of a simple pendulum depend only on the length of the string and the acceleration due to gravity

  • The period is independent of the mass

  • All simple pendula that are of equal length and are at the same location oscillate with the same period

Periodic Motion


Damped oscillations
Damped Oscillations from equilibrium (

  • In many real systems, nonconservative forces are present

    • This is no longer an ideal system (the type we have dealt with so far)

    • Friction is a common nonconservative force

  • In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped

Periodic Motion


Damped oscillations cont
Damped Oscillations, cont from equilibrium (

  • A graph for a damped oscillation

  • The amplitude decreases with time

  • The blue dashed lines represent the envelope of the motion

Periodic Motion


Damped oscillation example
Damped Oscillation, Example from equilibrium (

  • One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid

  • The retarding force can be expressed as R = - b v where b is a constant

    • b is called the damping coefficient

Periodic Motion


Damping oscillation example part 2
Damping Oscillation, Example Part 2 from equilibrium (

  • The restoring force is – kx

  • From Newton’s Second Law

    SFx = -k x – bvx = max

  • When the retarding force is small compared to the maximum restoring force we can determine the expression for x

    • This occurs when b is small

Periodic Motion


Damping oscillation example part 3
Damping Oscillation, Example, Part 3 from equilibrium (

  • The position can be described by

  • The angular frequency will be

Periodic Motion


Damping oscillation example summary
Damping Oscillation, Example Summary from equilibrium (

  • When the retarding force is small, the oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time

  • The motion ultimately ceases

  • Another form for the angular frequency

    where w0 is the angular frequency in the

    absence of the retarding force

Periodic Motion


Types of damping
Types of Damping from equilibrium (

  • is also called the natural frequency of the system

  • If Rmax = bvmax < kA, the system is said to be underdamped

  • When b reaches a critical value bc such that bc / 2 m = w0 , the system will not oscillate

    • The system is said to be critically damped

  • If Rmax = bvmax > kA and b/2m > w0, the system is said to be overdamped

Periodic Motion


Types of damping cont
Types of Damping, cont from equilibrium (

  • Graphs of position versus time for

    • (a) an underdamped oscillator

    • (b) a critically damped oscillator

    • (c) an overdamped oscillator

  • For critically damped and overdamped there is no angular frequency

Periodic Motion


Forced oscillations
Forced Oscillations from equilibrium (

  • It is possible to compensate for the loss of energy in a damped system by applying an external force

  • The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces

Periodic Motion


Forced oscillations 2
Forced Oscillations, 2 from equilibrium (

  • After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase

  • After a sufficiently long period of time, Edriving = Elost to internal

    • Then a steady-state condition is reached

    • The oscillations will proceed with constant amplitude

Periodic Motion


Forced oscillations 3
Forced Oscillations, 3 from equilibrium (

  • The amplitude of a driven oscillation is

    • w0 is the natural frequency of the undamped oscillator

Periodic Motion


Resonance
Resonance from equilibrium (

  • When the frequency of the driving force is near the natural frequency (w » w0) an increase in amplitude occurs

  • This dramatic increase in the amplitude is called resonance

  • The natural frequencyw0 is also called the resonance frequency of the system

Periodic Motion


Resonance1
Resonance from equilibrium (

  • At resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximum

    • The applied force and v are both proportional to sin (wt + f)

    • The power delivered is F . v

      • This is a maximum when F and v are in phase

Periodic Motion


Resonance2
Resonance from equilibrium (

  • Resonance (maximum peak) occurs when driving frequency equals the natural frequency

  • The amplitude increases with decreased damping

  • The curve broadens as the damping increases

  • The shape of the resonance curve depends on b

Periodic Motion


We are done

WE ARE DONE!!! from equilibrium (

Periodic Motion