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Chapter 1: Measurements & UnitsChapter 1: Measurements & UnitsChapter 2: 1D KinematicsChapter 2: 1D Kinematics
Thursday January 6thThursday January 6th•The International system of units (SI)
•Prefixes for SI units•Orders of magnitude•Changing units•Length, time and mass standards
•Motion in a straight line (1D Kinematics)•Average velocity and average speed•Instantaneous velocity and speed•Acceleration
Reading: pages 1 – 22 in the text book (Chs. 1 & Reading: pages 1 – 22 in the text book (Chs. 1 & 2)2)
Physics 2048: RemindersPhysics 2048: Reminders• First LON-CAPA assignment due MondayFirst LON-CAPA assignment due Monday
• First assignment mainly review (not all covered in lecture)First assignment mainly review (not all covered in lecture)• Very strict deadline of 11:59:59 pm on due dateVery strict deadline of 11:59:59 pm on due date
• First Mini-Exam next Thursday (Jan. 13First Mini-Exam next Thursday (Jan. 13thth))
• You must attend your first lab next week You must attend your first lab next week (M/W)(M/W)
Refer to Appendix A in Text Book
Trigonometry, Law of cosines, etc….
The International System of UnitsThe International System of UnitsSystème International (SI) d'Unités in FrenchSystème International (SI) d'Unités in French
a.k.a. metric or SI unitsa.k.a. metric or SI units•Scientists measure quantities through comparisons with standards.
•Every measured quantity has an associated unit.
•The important thing is to define sensible and practical "units" and "standards" that scientists everywhere can agree upon.(e.g. Rocket Scientists)
The International System of UnitsThe International System of UnitsSystème International (SI) d'Unités in FrenchSystème International (SI) d'Unités in French
a.k.a. metric or SI unitsa.k.a. metric or SI units
•Even though there exist an essentially infinite number of different physical quantities, we need no more than seven base units/quantities from which all others can be derived.In 1971, the 14th General Conference on In 1971, the 14th General Conference on
Weights and Measures picked these seven Weights and Measures picked these seven base quantities for the SI, or metric system.base quantities for the SI, or metric system.
In 1971, the 14th General Conference on In 1971, the 14th General Conference on Weights and Measures picked these seven Weights and Measures picked these seven base quantities for the SI, or metric system.base quantities for the SI, or metric system.
•In Mechanics, we really only need three of these base units (see table)
•In Thermodynamics, we need two more Temperature in kelvin (K) Quantity in moles (mol)
Prefixes for SI UnitsPrefixes for SI Units
33 560560 000000 000 m = 3.56 000 m = 3.56 × 10× 1099 m m
0.0000.000 000000 492 s = 4.92 × 10492 s = 4.92 × 10-7-7 s s
On LON-CAPA = 4.92*10^-7 sOn LON-CAPA = 4.92*10^-7 s
On your calculator (also LON-CAPA):On your calculator (also LON-CAPA):
4.92 × 104.92 × 10-7-7 = = 4.92 E-74.92 E-7 or or 4.92 4.92 -07-07
Prefixes (also works in LON-CAPA):Prefixes (also works in LON-CAPA):
5 5 × 10× 1099 watts = 5 gigawatts = 1 GW watts = 5 gigawatts = 1 GW
2 × 102 × 10-9-9 s = 2 nanoseconds = 2 ns s = 2 nanoseconds = 2 ns
Scientific notation:Scientific notation:
Orders of magnitudeOrders of magnitude
Changing unitsChanging unitsChain-link Chain-link conversion - an example:conversion - an example:
1 minute = 60 seconds
1 mintherefore 1
60 sec
60 secor 1
1 min
Note: this does not imply 60 = 1, or 1/60 = 1!Note: this does not imply 60 = 1, or 1/60 = 1!
2 min 2 min 1 60 s2 min
1 min
120 s
Conversion FactorsConversion Factors
Etc., etc., etc…….Etc., etc., etc…….
Resources at your fingertips…Resources at your fingertips…
Which of the following have been used as the Which of the following have been used as the standard for the unit of length corresponding to 1 standard for the unit of length corresponding to 1 meter?meter? a) One ten millionth of the distance from the North a) One ten millionth of the distance from the North pole to the equator.pole to the equator. b) The distance between two fine lines engraved b) The distance between two fine lines engraved near the ends of a platinum-iridium bar.near the ends of a platinum-iridium bar. c) 1 650 763.73 wavelengths of a particular orange-c) 1 650 763.73 wavelengths of a particular orange-red light emitted by atoms of krypton-86 (red light emitted by atoms of krypton-86 (8686Kr).Kr). d) The length of the path traveled by light in a d) The length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a vacuum during a time interval of 1/299 792 458 of a second.second.
Quiz #1
LengthLength
1 m = distance from N. pole to equator ten-million
1792: French established a new system of weights and measures
• Then, in the 1870s:
1 m = distance between fine lines on Pt-Ir bar
Accurate copies sent around the world
•Then, in 1960:
1 m = 1 650 763.73 × wavelength 86Kr (orange)•1983 until now (strict definition):
1 m = distance light travels in 1/(299 792
458) sec
TimeTime
•Length of the day
•Period of vibration of a quartz crystal
•Now we use Now we use atomic clocksatomic clocks
Some standards Some standards used through the used through the ages:ages:
1 second equivalent to 9 192 631 770 oscillations of the light emitted by a cesium-133 atom (133Cs) at a specified wavelength (adopted 1967)
United States Naval Observatory
MassMass
A second mass standard:A second mass standard:
•The The 1212C atom has been C atom has been assigned a mass of 12 assigned a mass of 12 atomic mass units (u)atomic mass units (u)
1 u = 1.6605402 × 101 u = 1.6605402 × 10-27-27 Kg Kg
•The masses of all other The masses of all other atoms are determined atoms are determined relative to relative to 1212CCNote: we measure "mass" in kilograms. Weight is something completely
different, which we measure in Newtons ( kgm/s2)
•Kilogram standard is a Pt-Ir cylinder in Paris
•Accurate copies have been sent around the world; the US version is housed in a vault at NIST
SummarySummary
-10 -8 -6 -4 -2 0 2 4 6 8 10x (m)
Motion in one-dimensionMotion in one-dimension•We will define the position of an object using the variable x, which measures the position of the object relative to some reference point (origin) along a straight line (x-axis).
Positive directionPositive direction
Negative directionNegative direction
-10
-8
-6
-4
-2
0
2
4
6
8
10x
(m)
Motion in one-dimensionMotion in one-dimension•We will define the position of an object using the variable x, which measures the position of the object relative to some reference point (origin) along a straight line (x-axis).
Pos
itiv
e di
rect
ion
Pos
itiv
e di
rect
ion
Neg
ativ
e di
rect
ion
Neg
ativ
e di
rect
ion
-10
-8
-6
-4
-2
0
2
4
6
8
10x
(m)
Motion in one-dimensionMotion in one-dimension•We will define the position of an object using the variable x, which measures the position of the object relative to some reference point (origin) along a straight line (x-axis).
•In general, x will depend on time t.
•We shall measure x in meters, and t in seconds, i.e. SI units.
•Although we will only consider one-dimensional motion here, we should not forget that x is really a vector. Thus, motion in the +x and -x directions corresponds to motion in opposite directions.
Pos
itiv
e di
rect
ion
Pos
itiv
e di
rect
ion
Neg
ativ
e di
rect
ion
Neg
ativ
e di
rect
ion
Graphing Graphing xx versus versus tt
-10
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-4
-2
0
2
4
6
8
10
t (s)54321
x (m
)
x(t)
-10
-8
-6
-4
-2
0
2
4
6
8
10
t (s)54321
x (m
)DisplacementDisplacement
DisplacementDisplacement xx::
xx = = xx22 xx11
final final initial position initial position
•Like Like xx, , xx is a is a vector quantityvector quantity•Its magnitude represents a distanceIts magnitude represents a distance•The sign of The sign of xx specifies direction specifies direction
x2
x1
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-6
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-2
0
2
4
6
8
10
t (s)54321
x (m
)Average velocityAverage velocity
xx = 4 = 4 mm ((1010 m)m) == 1414 mm
tt = 5 = 5 ss 00 == 55 ss
1
slopeof line
14m 2.8m.s
5s
avg
xv v
t
Average velocity and Average velocity and speedspeed
2 1
2 1avg
x xxv v
t t t
•Like displacement, Like displacement, vvavgavg is a vector is a vector
total distanceavgs s
t
• ssavgavg is not a vector - it lacks an algebraic sign is not a vector - it lacks an algebraic sign
•How do How do vvavgavg and and ssavgavg differ? differ?
Average speed Average speed ssavgavg::
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0
2
4
6
8
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t (s)54321
x (m
)12 1
2 1
( 10 ) ( 10 )0
6 0
x x m mv ms
t t s
1 118
63
v ms ms 1 1186
3v ms ms
1
total distance
18 186
6
st
m mms
s
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0
2
4
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t (s)54321
x (m
)Instantaneous velocity and Instantaneous velocity and
speedspeed0
lim local slopet
x dxv
t dt
v1
v2
v3
v3 > v1 > v2
Instantaneous speed = magnitude of vInstantaneous speed = magnitude of v
AccelerationAcceleration•An object is accelerating if its velocity is An object is accelerating if its velocity is changingchanging
Average acceleration Average acceleration aaavgavg::2 1
2 1avg
v vva a
t t t
Instantaneous acceleration Instantaneous acceleration aa::
2
20lim =
t
v dv d dx d xa
t dt dt dt dt
•This is the second derivative of the This is the second derivative of the xx vs. vs. tt graphgraph
•Like Like xx and and vv, acceleration is a vector, acceleration is a vector
•Note: direction of Note: direction of aa need not be the same as need not be the same as vv
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t (s)54321
x (m
)
AcceleratingAccelerating
aavv
DeceleratingDeceleratingaavv
x(t)
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t (s)54321
x (m
)
AcceleratingAccelerating
aavv
DeceleratingDeceleratingaavv
x(t)
v(t)
v3 > v1 > v2
-2
0
2
4
6
8
t (s)54321
v (m
/s)
AcceleratingAccelerating
aavv
DeceleratingDeceleratingaavv
v(t) a(t)
SummarizingSummarizing
Average velocity:Average velocity: 2 1
2 1avg
x xxv v
t t t
total distanceavgs s
t
local slope of versus graphdx
v x tdt
Displacement:Displacement: xx = = xx22 xx11
Instantaneous velocity:Instantaneous velocity:
Average speed:Average speed:
Instantaneous speed:Instantaneous speed: magnitude of magnitude of vv
SummarizingSummarizing
Average acceleration:Average acceleration:
Instantaneous acceleration:Instantaneous acceleration:
In addition:In addition:
2 1
2 1avg
v vva a
t t t
= local slope of versus graphdv
a v tdt
2
2= curvature of versus graph
d dx d xa x t
dt dt dt
SI units for SI units for aa are m/s are m/s22 or m.s or m.s22 (ft/min (ft/min22 also works) also works)
The sensation of accelerationThe sensation of acceleration•You will soon learn that, in order to accelerate an object, one must exert a net force on that object.
•Thus, when you accelerate, you experience stresses associated with the net force applied to your body.
•Therefore, your body is an accelerometer.•When you travel at a constant velocity, be it on a bicycle at 10 ms-1, or a supersonic aircraft at 1500 mph, you are not accelerating. Therefore, you do not feel the sensation of any net force, i.e. your body is not a speedometer.
•Acceleration is often expressed in terms of g units, where g (= 9.8 ms-1) is the magnitude of the acceleration of an object falling under gravity close to the earth's surface.
Constant acceleration: a special Constant acceleration: a special casecaseFor non-calculus derivations, see section 2-4 in text bookFor non-calculus derivations, see section 2-4 in text book
dv
a dv a dtdt
dv a dt v at C •To evaluate the constant of integration To evaluate the constant of integration CC, we let , we let v = v = vvoo at time at time tt = 0 = 0, i.e. , i.e. vvoo represents the initial velocity. represents the initial velocity.
2 7ov v at
•Taking the derivative of Eq. 2-7 (Taking the derivative of Eq. 2-7 (vvoo and and aa are are constants), we recover:constants), we recover: dv
adt
Further integrationFurther integration
dx
v dx v dtdt
( )dx v t dt •Note - in general, Note - in general, vv depends on depends on tt. However, we can . However, we can substitute substitute vv from Eq. 2-17: from Eq. 2-17:
212 'o odx v at dt x v t at C
•To evaluate the constant of integration To evaluate the constant of integration C'C', we let , we let x x = x= xoo at time at time tt = 0 = 0, i.e. , i.e. xxoo represents the initial position. represents the initial position.
212 2 10o ox x v t at
212o ox x v t at
ov v at
Equations of motion for constant Equations of motion for constant accelerationacceleration•Equations 2-11 and 2-15 contain 5 quantities: Equations 2-11 and 2-15 contain 5 quantities: aa, , tt, ,
vv, , vvoo and the displacement and the displacement xx - - xxoo. .
•Both equations contain Both equations contain aa, , tt and and vvoo..
•One can easily eliminate either One can easily eliminate either aa, , tt or or vvoo by solving by solving Eqs. 2-11 and 2-15 simultaneously.Eqs. 2-11 and 2-15 simultaneously.
Free fall accelerationFree fall acceleration
•If one eliminates the effects of air resistance, one If one eliminates the effects of air resistance, one finds that all objects accelerate downwards at the finds that all objects accelerate downwards at the same constant rate, regardless of their mass same constant rate, regardless of their mass (Galileo).(Galileo).
•That rate is called the free-fall acceleration That rate is called the free-fall acceleration gg..
•The value of The value of gg varies slightly with latitude, but for varies slightly with latitude, but for this course this course gg is taken to be is taken to be 9.8 ms9.8 ms-2-2 at the earth's at the earth's surface.surface.
•Thus, the equations of motion in table 2-1 apply Thus, the equations of motion in table 2-1 apply to free-fall near the earth's surface.to free-fall near the earth's surface.
•HRW substitute the coordinate HRW substitute the coordinate yy for for xx in problems in problems involving free-fall.involving free-fall.
•It is customary to consider It is customary to consider yy as increasing in the as increasing in the upward direction. Therefore, the acceleration upward direction. Therefore, the acceleration aa due due to gravity is in the negative to gravity is in the negative yy direction, i.e. direction, i.e. aa = -g = - = -g = -9.8 ms9.8 ms-2-2..
