PHYSICAL CONCEPTS

• Number issues

• Physical Quantities

• Force/Friction/Energy/Work, etc.

• Simple harmonic motion

• Vibration: Free and Forced

• Impedance

Scientific Notation

• number between 1.00 and 9.99 times 10 raised to some power

• E.G.,

1492 becomes

1.492 x 103 • 1.492 is called the COEFFICIENT

Multiplying numbers in Sci. Not.

• Multiply coefficients

• sum powers of 10

• E.G.

2.3 x 102 x 4x103

= (2.3 x 4) x 10(2+3)

= 9.2 x 105

Dividing in Sci. Not.

• Divide Coefficients

• Subtract Powers of 10

• Read More About Exponents in Appendix A

Quantities Come in 2 Flavors:

• Scalar Quantities– magnitude only

• Vectorial or Vector Quantities– magnitude AND direction

Scalar Quantities

• Have magnitude only

• Examples include Mass, Length, Volume

• Can be added or subtracted directly

Vector Quantities

• Have BOTH magnitude and direction

• Example: Velocity

• Combining Vectors is more complicated

Basic Units

• Length

• Time

• Mass

• (Charge)

Other Units may be derived:

• Area = Length x Length (or L2)

• Volume = L3

• Speed = Length/Time

• Acceleration = L/T2

Force: A push or a pull

• Force = Acceleration x mass

• Therefore Force = ML/T2

• MKS force unit is Newton = 1 kg m/s2

• cgs unit is dyne = 1 g cm/s2

Force and Elasticity

• Hooke’s Law:

• Force = (-)spring constant times displacement

• Stress = force per unit area (aka pressure)

• Strain = change in length

• Stress = Elasticity x Strain

Final Comment on Elasticity

• Compliance is the inverse of Stiffness

• Greater compliance yields more displacement per unit force

• Units: L/ML/T2

• (meters/newton, or cm/dyne)

Friction

• Energy converted into heat when molecules rub against each other.

• To move an object, the applied force must overcome friction.

• Effect of Friction is “Resistance”

Friction produces Resistance

• Resistance = ratio of Force to resulting velocity (R = f/v)

• measured in Ohms

• Acoustically, we talk about the influence of friction as DAMPING

Energy & Related Concepts

• WORK

• POTENTIAL AND KINETIC ENERGY

• POWER

WORK

• Force applied through a distance

• No motion--no work

• Work = force x distance = ML/T2 x L

• Units JOULE = 1 Newton Meter

• erg = 1 dyne cm

ENERGY COMES IN 2 FLAVORS

• Kinetic-- Energy of motion

• (Inertia can be thought of as the ability to store kinetic energy)

• Potential--Energy of position

• (Elasticity --ability to store potential energy)

POWER

• Rate at which work is done

• Work/Time

• Unit Watt = joule/second or 107 erg/sec

SIMPLE HARMONIC MOTION

• Vibration involves interplay of force, inertia, elasticity, and friction

• Applying a force displaces object

• Overcoming inertia

• Traveling away from rest until ?

Simple Harmonic Motion 2

• Why does object stop and then move back toward rest?

• Why doesn’t the object then stop at rest?

• Where is potential energy the greatest?

• Where is kinetic energy the greatest?

SHM 3

• Why does displacement decrease over time?

• RESISTANCE

• -- Energy is lost to HEAT through FRICTION

SHM 4

• Amplitude --Displacement

• Period-- Time taken to complete one cycle

• Frequency--Number of Cycles per Second

• Phase--Describing points in the Cycle

-1.2-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

11.2

1/1/00 2/1/00 3/1/00 4/1/00

TIME

DISPLACEMENT

A Waveform Shows Amplitude as a Function of Time

PEAK PEAK-TO-PEAK

AMPLITUDE MEASURES

• Instantaneous- amplitude at any given instant

• Peak

• Peak to Peak

• Root Mean Square--A way of getting average amplitude

• =Square root of Averaged Squared Amplitudes

Period and Frequency

• Frequency = 1/Period (in seconds)

• Units of Frequency = cycles per second or HERTZ

PHASE--Each cycle broken up into 360 degrees

• 0 degrees = 0 displacement and about to head positively

• 90 degrees = positive maximum

• 180 degrees=0 disp. About to head negatively

• 270 degrees= negative maximum

Phase Values Through a Cycle

-1.5-1

-0.50

0.51

1.5

1/1/00 2/1/00 3/1/00 4/1/00

Time

Displacement

90180 270

360

FREE VIBRATION

• Pendulum illustration represents FREE VIBRATION

• Force applied and object allowed to respond

• Frequency of Free Vibration =Resonant or Natural Freq.

• --determined by the object’s Mass and Stiffness

FORCED VIBRATION

• Force is applied back and forth

• Vibration occurs at the frequency of the applied force

• Object’s mass and stiffness determine amplitude of vibration

IMPEDANCE

• The opposition to vibration, or

• What, other than motion, happens to your applied force?

• That is what do you have to overcome?

Impedance has 3 components:

• Resistance: Energy lost to heat through friction

• Mass Reactance: Energy taken to overcome inertia

• Stiffness Reactance: Energy taken to overcome restoring force

Impedance and Frequency:

• Resistance is generally the same across frequency

• Reactance Components change with frequency

Reactance and Frequency:

• Mass reactance is greater at high frequencies

• --it’s harder to get massive objects to vibrate quickly

• Stiffness reactance is greater at low frequencies

• --it’s harder to get stiff objects to vibrate slowly

Mass and Stiffness Reactance

0

0.2

0.4

0.6

0.8

1

1.2

100 500 1000 4000

Frequency

Reactan

ce

Xm

XsRes

onan

t Fre

q.

At Resonant Frequency

• Mass and Stiffness Reactance Cancel

• Only opposition to vibration is Resistance

• In Forced Vibration, you get the most vibratory amplitude for amount of force applied