__NEWTON'S FIRST LAW OF MOTION__

if the resultant force acting on a particle is zero, the particle

will remain at rest (if originally at rest)

or

will move with constant speed in a straight line (if originally in motion)

__NEWTON'S SECOND LAW OF MOTION__

F = m a

__NEWTON'S THIRD LAW OF MOTION__

action and reaction are opposite and equal

__NEWTON'S LAW OF GRAVITATION__

F = G m_{1} m_{2} / r^{2}
where G = 6.673 X 10^{-11} N•m^{2}/kg^{2}

__THE LAW OF FLAT FRICTION__

F = μ_{s} N
where μ_{s} is coefficient of static friction

F = μ_{k} N
where μ_{k} is coefficient of kinetic friction

*dry* means *non-lubricated*

*wet* sometimes means *lubricated*

*wet* sometimes means *with water*

__TORQUE__

τ = F *l*

the unit is N•m

__WORK__

w_{cst force} = F x

the unit is J

note:

1 J = 1 N•m

__POWER__

P = dw/dt = F v

the unit is watt (W)

note:

1 W = 1 J/s

1 hp
= 550 ft•lb/s
≈ 746 W
≈ 0.746 kW ( ≈ ¾ kW )

__KINETIC ENERGY__

KE = ½ m v^{2}

the unit is J

__POTENTIAL ENERGY__

PE = m g y

the unit is J

__MOMENTUM__

P = m v

the unit is kg•m/s

__IMPULSE__

J = P_{2} - P_{1}
known as *impulse-momentum theorem*

the unit is kg•m/s

__LINEAR ( STRAIGHT-LINE ) MOTION__

x = x_{o} + v_{o} t + ½ a t^{2}

v = v_{o} + a t

v^{2} = v_{o}^{2} + 2 a ( x - x_{o} )

( x - x_{o} ) / t = ( v + v_{o} ) / 2

one of the application is of FREE FALL
and normally we define a = - g = - 9.8 m/s^{2}

__PROJECTILE MOTION__

A *projectile* is any body that is given an initial velocity & then follows a path determined entirely by the effects of gravitational acceleration & air resistence.
The path followed by a projectile is called *trajectory*.
We can treat the x-coordinate & y-coordinate separately,
in which a_{x} = 0 & a_{y} = - g.

__CIRCULAR MOTION__

*uniform circular motion* is the motion of a particle that moves in a circle with constant speed

*centripetal* means radially inward toward the center

*centrifugal* means radially outward from the center

*centripetal acceleration*,
a_{rad} = v_{t}^{2} / r

__ROTATIONAL MOTION__

ω is the *angular velocity*
( the unit is rad/s )

v_{t} = r ω

a_{t} = r α

a_{rad} = r ω^{2}

I = ∑ r_{i}^{2} m_{i}
( analogous to m )

τ = I α
( analogous to F = m a )

KE = ½ I ω^{2}
( analogous to KE = ½ m v^{2} )

w_{cst torque} = τ θ
( analogous to w_{cst force} = F x )

P = τ ω
( analogous to P = F v )

L = I ω
( analogous to P = m v )

θ = θ_{o} + ω_{o} t + ½ α t^{2}

ω = ω_{o} + α t

ω^{2} = ω_{o}^{2} + 2 α ( θ - θ_{o} )

( θ - θ_{o} ) / t = ( ω + ω_{o} ) / 2

__PERIODIC MOTION @ OSCILLATION__

ω is the *angular frequency*
( the unit is rad/s )

a *periodic motion @ oscillation* is a motion that repeats itself in a regular cycle

a *simple harmonic motion @ SHM* is a motion under the restoring force
( F = - k x ) that is directly proportional to the displacement from equilibrium and ω = 2 π f = ( k / m )^{½}

the position, velocity & acceleration of SHM are a sinusiodal functions of time

x = A cos ( ω t + Ø )

v = - A ω sin ( ω t + Ø )

a = - A ω^{2} cos ( ω t + Ø )

in a *angular SHM*,

τ = - k θ
analogous to F = - k x

ω = ( k / I )^{½}
analogous to ω = ( k / m )^{½}

__GUITAR__

below is the fret of guitar :

the ratio of distances of 2 consecutive frets to the bridge is 2^{1/12} ( ≈ 1.059463 )

theoretically, the 12^{th} fret should divide the string in 2 exact halves.

E = 329.6 Hz

B = 246.9 Hz

G, = 196 Hz

D, = 146.8 Hz

A, = 110 Hz

E,, = 82.4 Hz

note:

A' = 440 Hz ( known as *concert pitch* )

E = 329.6 Hz

E, = 329.6 / 2 = 164.8 Hz

E,,= 329.6 / 4 = 82.4 Hz

__MECHANICAL WAVE__

*mechanical waves* travel thru *medium* ( solid @ fluid )

*electromagnetic waves* can travel thru *empty space*

*wave speed*,
v = λ f

*angular frequency*,
ω = v k
where k is the *wave number @ propagation constant*,
k = 2 π / λ

y ( x , t ) = A sin ω ( t - x / v )

y ( x , t ) = A sin 2 π f ( t - x / v )

y ( x , t ) = A sin 2 π ( t / T - x / λ )

y ( x , t ) = A sin ( ω t - k x )

v = ( F / μ )^{½}
( transverse wave )

v = ( B / ρ )^{½}
( longitudinal wave - fluid )

v = ( Y / ρ )^{½}
( longitudinal wave - solid )

v = ( γ p / ρ )^{½}
( longitudinal wave - ideal gas )

v = ( γ R T / M )^{½}
( longitudinal wave - ideal gas )

__SUPERPOSITION & NORMAL MODES__

*principle of superposition* states that when 2 waves overlap, the actual displacement of any point of on the string, at any time, is obtained by adding the 2 displacements

*standing wave* is resultant wave based on principle of superposition

*reinforcement @ constructive interference* occurs when 2 @ more
*in phase* waves passing thru the same region at the same time

*cancellation @ destructive interference* occurs when 2 @ more
*out of phase* waves passing thru the same region at the same time

*resonance* occurs if force frequency = normal mode frequency @ force frequency = natural frequency

*normal mode of a string* ,
f_{n} = ( n v ) / 2 L = n f_{1}
( n = 1, 2, 3........... )

f_{1} = 1st normal mode = fundamental frequency

f_{2} = 2nd normal mode = 2nd harmonic = 1st overtone

f_{3} = 3rd normal mode = 3rd harmonic = 2nd overtone

f_{4} = 4th normal mode = 4th harmonic = 3rd overtone

*normal mode of air column (open pipe)* ,
f_{n} = ( n v ) / ( 2 L )
( n = 1, 2, 3........... )

*normal mode of air column (stopped pipe)* ,
f_{n} = ( n v ) / ( 4 L )
( n = 1, 3, 5........... )

terms:

node (never move at all)

antinode (amplitude is greatest)

pressure node = displacement antinode (pressure does not vary)

pressure antinode = displacement node (pressure variation is greatest)

__ACOUSTICS__

*acoustics* is a branch of physics which deals with sound

*sound* is longitudinal waves in air

*noise* is unwanted sound

sound can be classified as

* *ultrasonic* (with frequency above the range of human hearing)

* *audible* (with frequency within the range of human hearing , 20 Hz to 20 kHz)

* *infrasonic* (with frequency below the range of human hearing)

*intensity*,
I
=
½ ω B k A^{2}
=
p_{max}^{2} / ( 2 ρ v )
=
p_{max}^{2} / [ 2 ( ρ B )^{½} ]

*intensity level @ loudness*,
β = ( 10 dB ) log ( I / I_{o} )
where I_{o} = 10^{-12} W/m^{2}

*threshold of pain* :
120 dB & 1 W/m^{2} @ 1000 Hz

*threshold of hearing @ audibility* :
0 dB & 10^{-12} W/m^{2} @ 1000 Hz

noise induced hearing loss :

1) *temporary threshold shift*

10 minutes exposed to 120 dB will shift your threshold of hearing
from 0 dB to 28 dB

2) *permanent threshold shift*

10 years exposed to 92 dB will shift your threshold of hearing
from 0 dB to 28 dB

3) *acousic trauma*

due to explosion ... etc

*beats* are heard when 2 tones with slightly different frequencies
f_{1} and f_{2} are sounded together

*beat frequency*,
f_{beat} = f_{1} - f_{2}

the *pitch* of a musical tone is the quality that lets us classify
as high or low

it depends primarily on its frequency

eg. using piano

C = 262 Hz

C' = 262 X 2 = 524 Hz ( C' has higher pitch than C )

C" = 262 X 4 = 1048 Hz ( C" has higher pitch than C' )

__TEMPERATURE__

100^{o}C = 212^{o}F = 373 K

0^{o}C = 32^{o}F = 273 K

-273^{o}C = -460^{o}F = 0 K

note: *10 K* is read as *ten Kelvin* ; not *ten degrees Kelvin*

T_{K} = T_{C} + 273.15

T_{F} = ( 9 / 5 ) T_{C} + 32

T_{C} = ( 5 / 9 )( T_{F} - 32 )

__THERMAL EXPANSION__

thermal linear expansion, ΔL = α L_{o} ΔT

thermal volume expansion, ΔV = β V_{o} ΔT

example of coefficient of linear expansion :

aluminium = 2.4 X 10^{-5} K^{-1} @
(C^{o})^{-1}

brass = 2.0 X 10^{-5} K^{-1} @
(C^{o})^{-1}

copper = 1.7 X 10^{-5} K^{-1} @
(C^{o})^{-1}

steel = 1.2 X 10^{-5} K^{-1} @
(C^{o})^{-1}

example of coefficient of volume expansion :

aluminium = 7.2 X 10^{-5} K^{-1} @
(C^{o})^{-1}

brass = 6.0 X 10^{-5} K^{-1} @
(C^{o})^{-1}

copper = 5.1 X 10^{-5} K^{-1} @
(C^{o})^{-1}

steel = 3.6 X 10^{-5} K^{-1} @
(C^{o})^{-1}

return to main page