Chapter 15 Oscillatory Motion. Intro Periodic Motion- the motion of an object that regularly repeats There is special case of periodic motion in which.

Chapter 15

Oscillatory Motion

Intro

• Periodic Motion- the motion of an object that regularly repeats

• There is special case of periodic motion in which there is a force acting on an object proportional to the relative position of the object from equilibrium, and that force is directed towards equilibrium.

Intro

• A force such as this is called a restoring force, and results in Simple Harmonic Motion.

15.1 Motion of an Object Attached to a Spring

• 1st Example will be a block of mass m, attached to a spring of elastic constant k. • We’ve already studied the qualities of this system in Chapter 7, using Hooke’s Law and the work done by/on the spring.

15.1

• In this case, the spring provides the restoring force necessary for simple harmonic motion.

• If we apply Newton’s 2nd Law

kxFs

xmakx

m

kxax

15.1

• By definition, an object moves with SHM whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium.

• Note: The SHM motion of a spring also applies to a vertically aligned spring AFTER the spring is allowed to stretch to its new equilibrium.

• Quick Quiz p. 454

15.2 Mathematical Representation of SHM

• Remember that acceleration is the 2nd derivative of position with respect to time.

• We denote the term k/m with the symbol ω2 which gives

xm

k

dt

xd

2

2

xdt

xd 22

2

15.2

• We’re thinking integrate this twice to get the position function, but we haven’t seen an integration where a derivative contains the original function.

• This is called a second order differential equation. To solve this, we need a function x(t) that has a 2nd derivative that is negative of the original function, multiplied by ω2.

15.2

• The cosine and sine functions do just that. – Derivative of cos = -sin– Derivative of -sin = -cos

• The full solution to this 2nd order differential is

• Where A, ω, and φ are constants representing amplitude, angular frequency and phase angle respectively.

)cos()( tAtx

15.2

• Amplitude (A) - the maximum distance from equilibrium (in the positive or negative direction).

• Angular frequency (ω) – measures how rapidly oscillations are occurring (rad/s)– For a spring

• Phase Angle (φ) – uniquely determines the position of the object at t=0.

m

k

15.2

• The term (ωt + φ) represents the phase of oscillation.

• Keep in mind, the cosine function REPEATS every time ωt increases by 2π radians.

15.2

15.2

• Quick Quizzes p. 456

• Period (T) – the time for one oscillation.

• Frequency (f) – inverse of period, the number of oscillations per second. (Hz = s-1)

2

Tk

mT 2

Tf

1 f 2

m

kf

21

15.2

• Position function

• Velocity function

• Acceleration function

)cos()( 22

2

tAdt

xdta

)sin()( tAdt

dxtv

)cos()( tAtx

15.2

• Max values (occur when trig function = 1)– Position- Amplitude– Velocity- (passing through equilibrium)

– Acceleration- (at +/- Amp)

Am

kAv max

Am

kAa 2

max

15.2

• Plots of x(t), v(t), a(t)• When x(t) is at + amp– v(t) = 0– a(t) = -amax

• When x(t) = 0– v(t) = +/- vmax

– a(t) = 0

15.2

• Quick Quizzes p. 456

• Remember, the angular frequency (ω) is determined by (k/m)1/2

• A and φ are determined by conditions at t=0

• Examples 15.1-15.3

15.3 Energy of the Simple Harmonic Oscillator

• The kinetic energy of the SHO varies with time according to

• The Elastic Potential Energy of the SHO also varies with time according to

)(sin 222212

21 tAmmvK

)(cos22212

21 tkAkxU

15.3

• The total energy is therefore

• And simplifying…

• The total mechanical energy of a SHO is a constant of the motion and is proportional to the square of the amplitude.

)(cos)(sin 22221 ttkAUKE

221 kAE

15.3

• Plots of Kinetic and Potential Energy

15.3

• From the Energy equations we can determine velocity as a function of position.

• Solve for v

• See Figure 15.11 Pg 463• Example 15.4 pg 464

2212

212

21 kAkxmvUKE

22 xAv

15.4 Comparing SHM and UCM

• Simple Harmonic motion can easily be used to drive uniform circular motion.

• Piston- steam engine (train)internal combustion engine

15.4

• Simple Harmonic Motion along a straight line can be represented by the projectionof uniform circular motionalong the diameter of areference circle.

15.4

• Uniform circular motion is often considered to be a combination of two SHM’s. – One along the x axis.– One along the y axis.– The motions are 90o

or π/2 out of phase.

Quick Quiz p 467Example 15.5

15.5 Pendulums

• Simple Pendulum exhibits periodic motion, very close to a simple harmonic motion for small angles (<10o)

• The solution to the 2nd order differential for pendulums is

• Where

)cos(max t

l

g

15.5

• The period of oscillation is given as

• The period of a simple pendulum is independent of the mass attached, solely depending on the length and gravity.

• Quick Quizzes p. 469• Example 15.6

g

lT

22

15.5

• Physical Pendulum- an object that cannot be approximated as a point mass, oscillating through an axis that does not pass through its center of mass.

• The moment of inertia of the system must be accounted for

)cos(max t

I

mgd

15.5

• The period is given as

• Example 15.7 p 470

mgd

IT

22

15.5

• Torsional Pendulum- theoscillation involves rotation,when the support wire twists.• The object is returned to equilibrium by a restoring Torque.

15.5

• Where κ (kappa) is the torsion constant, often determined by applying a known torque to twist the wire through a measurable angle θ.

• Using Newton’s 2nd Law for Torques

• Rearranged 2

2

dt

dI

Idt

d

2

2

15.5

• We see that this is the same form as our 2nd Order differential for the mass/spring system, with κ/I representing ω2.

• There is no small angle restriction for the torsional pendulum, so long as the elastic limit of the wire is not exceeded.

I

IT

2

15.6 Damped Oscillations

• So far we have discussed oscillations in ideal systems, with no loss of energy.

• In real life, energy is lost mostly to internal forms, diminishing the total ME with each oscillation.

• An oscillator whose energy diminishes with time is said to be Damped.

15.6

• One common example is when the oscillating object moves through a fluid (air), creating a resistive force as discussed in Ch 6.

• At low speeds the resistive or retarding force is proportional to the velocity as

• Where b is called the damping constant. vR b

15.6

• Applying Newton’s 2nd Law

• Again a 2nd Order Differential, the solution is a little more complicated.

• Where

2

2

dt

xdm

dt

dxbkx

)cos()( 2 tAetx tmb

2

2

m

b

m

k

15.6

• Recognize that (k/m)1/2 would be the angular

frequency in the absence of damping, called the “Natural Angular Frequency” (ωo)

• The motion occurs within the restricting envelope of the exponential decay.

22

2

m

bo

15.6

15.6

• Damping falls into three categories based on the value of b. – Underdamped- when the value of bvmax < kA• As the value of b increases the ampltitude of the

oscillations decreases rapidly.

– Critically Damped- when b reaches the critical value bc such that

• When the object is released it will approach but never pass through equilibrium

oc

m

b 2

15.6

– Overdamped- when the value of bvmax > kA and

• The fluid is so viscous that there is no oscillation, the object eventually returns to equilibrium.

oc

m

b 2

15.6

• Quick Quiz p. 472

15.7 Forced Oscillations

• Because most oscillators experience damping, an external force can be used to do positive work on the system compensating for the lossed energy.

• Often times the value of the force varies with time, but the system will often reach a steady state where the energy gained/lost is equal.

15.7

• The system will oscillate according to

• With a constant Amplitude given by)cos( tAx

2

222

mb

mFA

o

o

15.7

• Notice when ω≈ωo, the value of A will increase significantly.

• Essentially when the Forcing frequency matches the natural frequency oscillations intensify. – Examples- • Loma Prieta “World Series” Earthquake• Tacoma Narrows Bridge