Periodic MotionPeriodic Motion

What is periodic motion?What is periodic motion?

When a When a vibrationvibration or or oscillationoscillation repeats itself over and over the repeats itself over and over the motion is said to be motion is said to be periodicperiodic. . Most objects vibrate briefly when given Most objects vibrate briefly when given

an impulsean impulse Electrical oscillations occur in TV’s and Electrical oscillations occur in TV’s and

radio’s, atoms vibrate around a fixed radio’s, atoms vibrate around a fixed spotspot

TerminologyTerminology Oscillation or Vibration Oscillation or Vibration – a motion that repeats – a motion that repeats

itself with no net displacement.itself with no net displacement. Equilibrium Position Equilibrium Position – the point that the object – the point that the object

oscillates around. Also known as the rest oscillates around. Also known as the rest position.position.

DisplacementDisplacement – how far the mass is from the – how far the mass is from the equilibrium point (x)equilibrium point (x)

Maximum displacement Maximum displacement – how far the mass – how far the mass moves from the equilibrium position. (xmoves from the equilibrium position. (xmax max

occurs at A)occurs at A) AmplitudeAmplitude (A) – the distance from the (A) – the distance from the

equilibrium point to the maximum displacement.equilibrium point to the maximum displacement. CycleCycle – a complete to and fro motion. – a complete to and fro motion.

TerminologyTerminology PeriodPeriod ( (TT) – the time needed to ) – the time needed to

complete one cycle. (units – complete one cycle. (units – seconds)seconds)

FrequencyFrequency ( (ff) – the number of cycles ) – the number of cycles completed in one second. Units are completed in one second. Units are Hertz. (Hz = 1/s = sHertz. (Hz = 1/s = s-1-1))

FormulaFormula f f =1/T =1/T T=1/T=1/ff

Simple Harmonic MotionSimple Harmonic Motion Simple Harmonic Motion Simple Harmonic Motion is any is any

motion in which the restoring force is motion in which the restoring force is proportional to displacement.proportional to displacement. Examples:Examples:

An acrobat swinging on a trapezeAn acrobat swinging on a trapeze Child on a playground swingChild on a playground swing Pendulum of a clock or metronomePendulum of a clock or metronome Mass on the end of a springMass on the end of a spring

Restoring force Restoring force – the force that – the force that pushes or pulls the mass back to pushes or pulls the mass back to equilibrium.equilibrium.

Hooke’s LawHooke’s Law In 1678 Robert Hooke found that most In 1678 Robert Hooke found that most

mass-spring systems obey a simple mass-spring systems obey a simple relationship between force and relationship between force and displacement for small displacements.displacement for small displacements.

This is a restoring force.This is a restoring force.

FFSpringSpring = F = FElasticElastic = – kx = – kx The force, The force, FF, is negative because it is , is negative because it is

always a restoring force, pulling or pushing always a restoring force, pulling or pushing the opposite direction of the displacement.the opposite direction of the displacement.

The Applied Force is in the opposite The Applied Force is in the opposite direction of the Spring Force.direction of the Spring Force.

Spring Constant “k”Spring Constant “k” The value of the constant measures The value of the constant measures

the ‘stiffness’ of the spring. the ‘stiffness’ of the spring. The larger the value, the stiffer the The larger the value, the stiffer the

springspring Unit for the spring constant, k, is Unit for the spring constant, k, is

N/m.N/m.

Period of a SpringPeriod of a Spring Period of a spring is given by the following Period of a spring is given by the following

equation:equation: m = mass in kgm = mass in kg k = spring constant in N/mk = spring constant in N/m

What happens to the period as mass What happens to the period as mass increases?increases?

What happens to the period as the spring What happens to the period as the spring constant increases?constant increases?

What is the frequency equation?What is the frequency equation?

k

mT 2

mk

f21

Elastic Potential EnergyElastic Potential Energy At maximum displacement the potential At maximum displacement the potential

energy is at its maximum and kinetic is at energy is at its maximum and kinetic is at its minimum.its minimum.

At the equilibrium position the potential is At the equilibrium position the potential is at its minimum and kinetic is at its at its minimum and kinetic is at its maximum.maximum.

UUS S = U= USpringSpring = ½ k x = ½ k x2 2

KE = ½ m vKE = ½ m v22

Mass on a SpringMass on a SpringHORIZONTAL MOTIONHORIZONTAL MOTION

If the object is in If the object is in MOTIONMOTION then at the then at the equilibrium position (x=0) the velocity equilibrium position (x=0) the velocity is at the maximum.is at the maximum.

At the maximum displacement, spring At the maximum displacement, spring force and acceleration reaches a force and acceleration reaches a maximum and the velocity is zero.maximum and the velocity is zero. Known as a turning point.Known as a turning point. The acceleration is in the opposite The acceleration is in the opposite

direction of the motion.direction of the motion.

x0

A

A

E = K + UE = 1/2mv2 + 1/2kx2

Maximum displacement is A = x for the example below.

E = 1/2kA2

E = 1/2mv02

E = 1/2kA2

Conservation of energy Conservation of energy reviewreview

Remember that all energy is conserved Remember that all energy is conserved The energy just changes from one form to an The energy just changes from one form to an

other.other. Initial Energy = Final EnergyInitial Energy = Final Energy

KKii + U + Uii = K = Kff +PU +PUff

2 2 2 2f

2 2f

2f

The spring is release from "A";

what is the maximum speed ?

1 1 1 1m 0 k A m v k 0

2 2 2 2k

v Am

k kv A A

m m

The example is for a horizontal spring system.Solving for “vf” as for a spring released at “A” and finding the maximum velocity as the spring passes through the equilibrium position.

Finding the velocity in general depends on the Finding the velocity in general depends on the original amplitude and the location in the cycle, x.original amplitude and the location in the cycle, x.

2 2 2

2 22 2 2

1 1/ 2 1 22

1 2 1 2

1 2

SKE U E

mv kx kA

kA kx k kv A x

m m m

2

2202

222 11

Ax

vAx

Amk

v

2

2

0 1Ax

vv

Potential Energy in a Spring Potential Energy in a Spring ExampleExample

A spring with a force constant of 5.2 N/m has a relaxed length of 2.45 m. When a mass is attached to the end of the spring and allowed to come to rest, the vertical length of the spring is 3.57 m. Calculate the elastic potential energy stored in the spring.

Spring Constant in a Spring Example

A mass of 0.30 kg is attached to a spring and is set into vibration with a period of 0.24 s. What is the spring constant of the spring?

Period and Frequency in a Spring Example

A spring of spring constant 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: 2.3 kg 15 g 1.9 kg

Simple Graph of SHMSimple Graph of SHM

cos tx t A

Cosine GraphCosine Graph

This means the position is a function of time.This means the position is a function of time.

Graph of Unit CircleGraph of Unit Circle

=00

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

Time

Th

eta

x0

=/2-1.20

-0.70

-0.20

0.30

0.80

1.30

0 1 2 3 4 5 6 7 8

Time

Th

eta

x0

= -1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 1 2 3 4 5 6 7 8

Time

Th

eta

x0

=3-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 1 2 3 4 5 6 7 8

Time

Th

eta

x0

Notice that the radius of the circle Notice that the radius of the circle equals the amplitude of the spring.equals the amplitude of the spring.

=2

Cosine Graph

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 2 4 6 8 10

Time

Th

eta

x0

Amplitude is independent of the Amplitude is independent of the period!!!period!!!

The maximum velocity is equal to the The maximum velocity is equal to the path length of the circle (2path length of the circle (2r) divided r) divided by time.by time.

AfT

A

t

rv

222

0 k

m

v

A

mvkA

0

20

2

21

21

k

mT 2

Is a pendulum simple harmonic Is a pendulum simple harmonic motion?motion?

Simple pendulum is a mass on the end Simple pendulum is a mass on the end of a string. of a string.

The mass is called a “BOB”.The mass is called a “BOB”. Assume the mass is concentrated at a Assume the mass is concentrated at a

point.point. Neglect air resistance and friction.Neglect air resistance and friction.

Simple PendulumSimple Pendulum

The restoring force is a component of The restoring force is a component of the bob’s weight (-mg sin the bob’s weight (-mg sin ).).

If the restoring force is proportional If the restoring force is proportional to the displacement the pendulum’s to the displacement the pendulum’s motion is simple harmonic.motion is simple harmonic.

There are two forces acting on the There are two forces acting on the pendulum:pendulum: The tension in the stringThe tension in the string The weight of the bobThe weight of the bob

Ftension

Weight

Ftension

Weight

WeightyWeightx

Motion of a Pendulum

Restoring forceRestoring force As the pendulum is pulled back the x As the pendulum is pulled back the x

component of the weight gets larger and component of the weight gets larger and the y component gets smaller.the y component gets smaller.

Therefore the greater the displacement Therefore the greater the displacement the larger the restoring forcethe larger the restoring force

For small displacements the pendulums For small displacements the pendulums motion is simple harmonic.motion is simple harmonic.

Energy of a PendulumEnergy of a Pendulum Energy is conserved.Energy is conserved. At maximum displacement: Velocity At maximum displacement: Velocity

is zero, acceleration is largest, is zero, acceleration is largest, Energy is all potential.Energy is all potential.

At equilibrium: Velocity is the At equilibrium: Velocity is the largest, acceleration is zero, Energy largest, acceleration is zero, Energy is all kinetic.is all kinetic.

Amplitude, Period, and Amplitude, Period, and FrequencyFrequency

The time it takes for a pendulum to The time it takes for a pendulum to swing from one side to the other and swing from one side to the other and back again is one back again is one periodperiod..

The number of complete cycles in The number of complete cycles in one second is the one second is the frequencyfrequency..

Small Angle NotesSmall Angle Notes

Extra NotesExtra Notes

Figure 14.19Figure 14.19

Figure 14.19BFigure 14.19B

Figure 14.19AFigure 14.19A

What is this length here?

Answer is l cos

What is this length here?

Answer is l (1 - cos

l sin

Trig of a PendulumTrig of a Pendulum

mg

FT

mg cos

mg sin

Restoring force = -mg sinFor small angles sin = Using x = L gets:F = -(mg/L)xThis is similar to Hooke’s Law with k = mg/LUsing the equations derived for a spring :

x

L

g

L

Lmgm

k

m 222

Note that the period of Pendulum does NOT depend on the mass of the bob!!

Period of a PendulumPeriod of a Pendulum Depends on the length and free fall Depends on the length and free fall

acceleration.acceleration. For small amplitudes the period For small amplitudes the period

DOES NOT depend on the amplitude.DOES NOT depend on the amplitude.

gL

T 2

Damped Harmonic MotionDamped Harmonic Motion The amplitude of any real oscillating The amplitude of any real oscillating

spring slowly decreases. This is spring slowly decreases. This is damped harmonic motiondamped harmonic motion

Damping is due to friction and airDamping is due to friction and air

Forced Vibrations/ Forced Vibrations/ ResonanceResonance When a system is set in motion then left When a system is set in motion then left

alone it vibrates at its natural frequency alone it vibrates at its natural frequency (f(f00))

When an outside force is constantly When an outside force is constantly applied it creates forced vibrationsapplied it creates forced vibrations

The amplitude of the forced vibration The amplitude of the forced vibration depends on the difference between f and depends on the difference between f and ff00..

Now with x(t) find v(t) and Now with x(t) find v(t) and a(t)a(t)

2

2

x

( ) sin

cos t

take the derivative of w.r.t. time

cos t sin t

take the derivative of w.r.t. time

sin t c

v

( )

t

t

t

s

s

o

o

c

x t A

dx dA A

dt dt

dv dA A

dt

a A

d

t

A

t

v t