Simple Harmonic Motion

OscillationsOscillations

• Motion is repetitive (Motion is repetitive (PeriodicPeriodic) and the oscillating ) and the oscillating body moves back and forth around an equilibrium body moves back and forth around an equilibrium position. position.

• PeriodPeriod: The time required for one full oscillation : The time required for one full oscillation • We will focus on constant periods… We will focus on constant periods…

• What are some examples of oscillating What are some examples of oscillating bodies/systems?bodies/systems?

Periodic Motion• Objects that move back and forth periodically are described as oscillating.

• These objects move past an equilibrium position, O (where the body would rest if a force were not applied) and their displacement from this position changes with time.

• If the time period is independent of the maximum displacement, the motion is isochronous.

O

A time trace is a graph showing the variation of displacement against time for an oscillating body.

Demo: Producing a time-trace of a mass on a spring.

Distance sensor connected to computer

E.g.-Oscillating pendulums, watch springs or atoms can all be used to measure time

Properties of oscillating bodies

Amplitude (x0): The maximum displacement (in m) from the equilibrium position (Note that this can reduce over time due to damping).

Cycle: One complete oscillation of the body.

Period (T): The time (in s) for one complete cycle.

Frequency (f): The number of complete cycles made per second (in Hertz or s-1). (Note: f = 1 / T)

Angular frequency (ω): Also called angular speed, in circular motion this is a measure of the rate of rotation. In periodic motion it is a constant (with units s-1 or rad s-1) given by the formula…

ω = 2π = 2πf T

Q.

Calculate the angular speed of the hour hand of an analogue watch (in radians per second).

Angle in one hour = 2π radians

Time for one revolution = 60 x 60 x 12 = 43200s

ω = 2π = 1.45 x 10-3 rad s-1

T

Simple Harmonic Motion (SHM)

Consider this example

Hooke’s LawHooke’s Law

• What is the relationship between the displacement of What is the relationship between the displacement of the spring and the force applied to the spring? the spring and the force applied to the spring?

• What characteristics of the spring will affect this? What characteristics of the spring will affect this? • Hooke’s LawHooke’s Law:: up to its elastic limit, a spring will up to its elastic limit, a spring will

experience a displacement that is proportional to the experience a displacement that is proportional to the force applied to the spring.force applied to the spring.

More on springs… • Restoring ForceRestoring Force: :

• the tension in the spring that works to bring a the tension in the spring that works to bring a mass back to its rest position (equilibrium mass back to its rest position (equilibrium position). position).

• The restoring force is the reaction force to The restoring force is the reaction force to any applied force (i.e. the weight of any applied force (i.e. the weight of something hanging from the spring) something hanging from the spring)

• This is the force that causes a mass to accelerate This is the force that causes a mass to accelerate around its rest position when it has experienced around its rest position when it has experienced a displacement away from equilibrium.a displacement away from equilibrium.

Simple Harmonic Motion

• A special case of periodic oscillations that can be described by analyzing the forces involved in the motion

• Hooke’s law can be rewritten as… ma = -k x• We are going to define • angular frequency as: ω = √ (k /

m)• Angular frequency has units of Hertz (s-1)

Defining relation for SHM:Defining relation for SHM:a = -a = -ωω22 x x

• ““Simple harmonic motion takes place when a particle Simple harmonic motion takes place when a particle that is disturbed away from its fixed equilibrium that is disturbed away from its fixed equilibrium position experiences an acceleration that is proportional position experiences an acceleration that is proportional and opposite to its displacement” and opposite to its displacement” (from the IB Physics (from the IB Physics text by Tsokos) text by Tsokos) Two requirements for SHM: Two requirements for SHM:

• Must have a fixed equilibrium point Must have a fixed equilibrium point • Acceleration, when displaced, must be Acceleration, when displaced, must be

proportional to the amount of displacementproportional to the amount of displacement

SHM--mathematics

• By using calculus, the defining relationship becomes one that we can put in terms of the angular frequency, the time that has passed, the amplitude of the displacement, and the “phase shift”

• A = amplitude x = A cos (ωt + Φ)• t = time • Φ = phase shift • ω = angular frequency

• Amplitude:Amplitude: the maximum displacement away from the the maximum displacement away from the equilibrium (rest) position. equilibrium (rest) position. • This occursThis occurs when the value of the cosine function is when the value of the cosine function is

equal to 1 equal to 1 • ΦΦ = Phase Shift: = Phase Shift: recorded in radians; gives an indication recorded in radians; gives an indication

of the displacement at t = 0 s. of the displacement at t = 0 s. • See diagram on next slide… See diagram on next slide…

• Phase difference:Phase difference: ΔΦ = |Φ1 – Φ2|

Simple Pendulum…Simple Pendulum…

• Is this simple harmonic motion? How do you Is this simple harmonic motion? How do you know? know? NO!NO! It is It is notnot SHM—the acceleration and the SHM—the acceleration and the

displacement are not proportional to each other displacement are not proportional to each other Period of a pendulum can be found with:Period of a pendulum can be found with:

T = 2π √(L/g)

Q1

Sketch a graph of acceleration against displacement for the oscillating mass shown (take upwards as positive.

o

displacement

x

a

Q2

Consider this duck, oscillating with SHM…

Where is…

i. Displacement at a maximum?

ii. Displacement zero?

iii. Velocity at a maximum?

iv. Velocity zero?

v. Acceleration at a maximum?

vi. Acceleration zero?

A and E

C

C

A and E

A and E

C

Damped Oscillations

• Any oscillations that are taking place with the presence of one or more resistance forces, such as friction and air resistance.

• As a result of resistance forces, oscillations will eventually stop as the energy of its motion is transformed into other forms of energy—primarily thermal energy of both the environment and the system.

Types of damping effects:

• Under-damping: • Small resistance forces • Oscillations continue, but with a slightly smaller

frequency than if force was not there • Amplitude gradually reduces until it approaches 0

and the oscillations stop • The larger the damping force, the longer the period

and the faster it will stop oscillating

Types of damping effects:

• Critical Damping: • The system returns to its equilibrium state

as fast as possible without any oscillations

• Over-damping: • The system returns to equilibrium without

oscillations, but much slower than in critical damping

Graphical comparison of damping effects

Oscillations and Waves

Energy Changes During Simple Harmonic Motion

Energy in SHMEnergy-time graphs

velo

city

KE

PE

Totalener

gy

Note: For a spring-mass system:KE = ½ mv2 KE is zero when v = 0PE = ½ kx2 PE is zero when x = 0 (i.e. at vmax)

Energy–displacement graphs

energy

displacement

+xo-xo

KE

PE

Total

Note: For a spring-mass system:KE = ½ mv2 KE is zero when v = 0 (i.e. at xo)PE = ½ kx2 PE is zero when x = 0

Kinetic energy in SHM

We know that the velocity at any time is given by…

v = ω √ (xo2 – x2)

So if Ek = ½ mv2 then kinetic energy at an instant is given by…

Ek = ½ mω2 (xo2 – x2)

Potential energy in SHM

If a = - ω2x

then the average force applied trying to pull the object back to the equilibrium position as it moves away from the equilibrium position is…

F = - ½ mω2x

Work done by this force must equal the PE it gains (e.g in the springs being stretched). Thus..

Ep = ½ mω2x2

Total Energy in SHM

Clearly if we add the formulae for KE and PE in SHM we arrive at a formula for total energy in SHM:

ET = ½ mω2xo2

Summary:

Ek = ½ mω2 (xo2 – x2)

Ep = ½ mω2x2

ET = ½ mω2xo2

SOURCES

teacherweb.com/CA/.../Berg/SimpleHarmonicMotionweb.ppt

sjhs-ib-physics.wikispaces.com/file/view/15+Kinematics+of+SHM.ppt