CHAPTER 8
Time and Time-Related Parameters
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8-1
Fundamental Dimensions
• Length
• Time and time-related parameters
• Mass
• Temperature
• Electric current
• Amount of substance
• Luminous intensity
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8-2
Outline
In this chapter we will
• Investigate the role of time as a fundamental
dimension
• Learn about time-related parameters in
engineering applications
 frequency
 period
 traffic flow
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8-3
Outline (continued)
• Learn about engineering variables involving
length & time
 linear velocity
 linear acceleration
 volume flow rate
 rotational motion
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8-4
Learning Engineering Fundamental Concepts and Design
Variables from Fundamental Dimensions – Time
Fundamental dimensions and how they are used in defining
variables that are used in engineering analysis and design
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8-5
Why Is Time Important?
• Time is an important parameter in describing motion
• We live in a dynamic world where everything is in
constant motion
• Think about some of the questions frequently asked in
our daily lives
 how old are you?
 how long does it take to cook this food?
 how late is the store open?
 how long is your vacation?
 how long does it take to go from here to there?
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8-6
Role of Time in Our Lives
How much time do we have in our lives?
• Average life span of a person is about 75
years (657,450 hours  660,000 hours)
• We spend about 220,000 hours sleeping
• For an average 18 years old freshman, you
have about 330,000 waking hours available to
you if you live to the age of 75 years
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8-7
What Is Time?
Time is one of the seven fundamental or base
dimensions that we use to properly express
events in our surroundings
Base SI Unit – second
second – duration of 9,192,631,770 periods of the
radiation corresponding to the transition between
the 2 hyperfine levels of the ground state of cesium
133 atom
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8-8
Units or Divisions of Time
Even more
divisions micro second
nano second
• second
• month
• minute
• year
• hour
• decade
• day
• century
• week
• millennia
On your birthday, when someone asks you how old you
are, do you say: I am 170,000 hours old or I am 19 years
old?
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8-9
Longitudes and Latitudes
• Longitudes
 Typically, on maps, the earth is
divided into 360 circular arcs that are
equally spaced from east to west.
These arcs are called longitudes and
the zero longitude passes through
Greenwich, England.
• Latitudes
 Measure the angles formed by the
lines connecting the center of the
earth to the specific locations on the
earth surface and the equatorial
plane
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8-10
Time Zones
• Earth rotates about the north-south axis and
completes one revolution (360o) per 24 hours
or per day
• Every 15-degrees longitude corresponds to 1
hour
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8-11
Standard Time Zones
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8-12
Time Zone and Daylight Saving Time
• Why do we need time zones?
th century railroad companies
 During the 19
realized a need for standardizing their
schedules
• Why do we have daylight saving time?
 It was originally put into place to save fuel
during hard times such as World War I
and World War II.
 It also encourages people to engage in
more outdoor activities
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8-13
Time’s Role in Engineering Applications
• Steady state
 When the value of a physical quantity under
investigation does not change with time (Any
Examples?)
• Transient (unsteady) state
 When the value of a physical quantity under
investigation changes with time (Any Examples?)
• Some Engineering Applications
 Electronic equipment, biomedical applications,
combustion, materials testing, food processing,
material processing, heating & cooling applications,
wind & earthquake engineering
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8-14
Periods and Frequencies
• Period
 the time that it takes for an event to repeat itself
• it takes about 365 days for your birthday to come around
• Frequency
 inverse of period
• your birthday comes once a year
• Role of periods & frequencies
 important design parameters in: a structure’s response
to wind and earthquake, a mechanical system behavior
such piston inside a car’s engine cylinder, electrical &
electronic systems
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8-15
Periods and Frequencies
• period
1
m
T
 2
fn
k
in seconds
• frequency
A simple spring-mass
system
1
fn 
2
k
m
in Hz, cycles
per second
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8-16
Examples of Frequencies of Various
Electrical and Electronic Systems
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8-17
Example 8.1 – Period and Frequency
Given: a simple spring-mass system
Find: the natural frequency of the system
Solution:
1
fn 
2
k
1

m 2
5000 N/m
 8Hz
2 kg
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8-18
Traffic Flow
Flow of traffic
3600n
q
T
q = number of vehicles per hour
n = number of vehicles passing through a known
location in time T
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8-19
Traffic Flow
Traffic density k – number of
cars on a stretch of highway
1000n
d
k  number of vehicles per kilometer
k 
n  number of vehicles
d  the stretch of highway in meters
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8-20
Traffic Flow
relationship between flow
of traffic, density, and
average speed
q  ku
q  flow of traffic (number of vehicles per hour)
k  density (number of vehicles per kilometer)
u  average speed (kilometer s per hour)
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8-21
Example 8.2 – Traffic Flow
Given: the traffic flow equation
q  ku
Find: show the equation is dimensionally homogeneous
Solution:
 vehicles 
 vehicles   kilometer 
q
  k
u 

 hour 
 kilometer   hour 
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8-22
Time-Related Variables – Linear
Velocities
• Provide a measure of how fast an object moves
• Examples?
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8-23
Linear Velocity
2 fundamental
dimensions:
length and time
• Average speed
change in the position of the moving object
time
distance traveled

time
average speed 
• Instantaneous speed – actual speed at any given instant
• Instantaneous velocity – actual speed and direction at
any given instant
• Units: m/s, km/hr, ft/s, miles/hr
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8-24
Examples of Some Speeds
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8-25
Linear Acceleration
• Provides a measure of how velocity changes with time
change in velocit y
average accelerati on 
time
2 fundamental
dimensions:
length and time
• Instantaneous acceleration – actual acceleration at a
given instant
• Since velocity is a vector, acceleration is a vector
• Units: m/s2, ft/s2
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8-26
Acceleration
Can we have an acceleration without
a change in speed?
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8-27
Acceleration Due to Gravity
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8-28
Speed, Acceleration, and Distance
speed, acceleration, and distance traveled by a falling object,
neglecting the air resistance
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8-29
Example 8.3 – Acceleration, Speed,
and Distance
Given: the variation in the speed of a
car as shown
Find: the total distance traveled by
the car and its average speed over
this distance
Solution:
Between 0 and 15 seconds, the average speed is 50 mph
 km  1 h  1000 m 
d1  time average speed   15 s  50


  208.3 m
h  3600 s  1 km 

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8-30
Example 8.3 – Acceleration, Speed,
and Distance
Solution (continued):
Between 15 and 1815 seconds (that is a duration of
30 minutes), average speed is 100 mph
km  1 h  1000 m 

d 2  1800 s 100


  50000 m
h
3600
s
1
km




Between 1815 and 1825 seconds, average speed is 50 mph
 km  1 h  1000 m 
d 3  10 s  50


  138.9 m
h  3600 s  1 km 

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8-31
Example 8.3 – Acceleration, Speed,
and Distance
Solution (continued):
Total distance traveled by the car is
d  d1  d 2  d3  208.3  50000  138.9  50347.2 m  50.3472 km
Average speed of the car for the entire duration of travel is
Vaverage
distance traveled 50347.2


 27.56 m/s  99.2 km/h
time
1825
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8-32
Volume Flow Rate
2 fundamental
dimensions:
length and time
• Volume flow rate
volume
volume flow rate 
time
• Units: length3 per unit time
3
3
3
 m /s, ft /s, gallons/day, m /hr
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8-33
Volume Flow Rate
more in fluid
mechanics &
heat transfer
• Examples
 water consumptions in gallons/day
3
 natural gas used in m /hr
3
 air supply in m /s
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8-34
Example 8.4 – Flow Rate
Given: the piping system shown--water flows from a 12inch pipe into a 6-inch pipe steadily
Find: the volume flow rate of water in ft3/s,
gallons/minute, and liters/second; the average speed of
water in the 6-in-diameter pipe
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8-35
Example 8.4 – Flow Rate
Solution:
Volume flow rate, Q
Q  average velocity cross - sectional area of flow 
 ft   
2
Q   5  1 ft   3.926 ft 3 /s
 s  4 
 7.48 gal  60 s 
3
Q  3.926 ft /s 

  1762 gpm
3
 1 ft  1 min 




 28.31 L 
Q  3.926 ft 3 /s 
  111.2 L/s
3
 1 ft 
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8-36
Example 8.4 – Flow Rate
Solution (continued):
the average speed of water in 6-in pipe,
Q  average speed cross - sectional area of flow 
 
2
3.926 ft /s  average speed  0.5 ft 
4
average speed  20 ft/s
3
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8-37
Angular (Rotational) Speed
• Measures the change of angular
position over time
• Angular (rotational) speed


in radians per second
t
• another common unit: rpm
(revolutions per minute)
• 1 revolution = 2 radians
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8-38
Angular Motion Examples
• Shafts
• Wheels
• Gears
• Drills
• Fan or pump impellers
• DVD or hard drives
• Helicopter blades
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8-39
Angular Speed and Linear Speed
S  r
and dividing both sides by t
S

r
t
t

with  
t
Then we have :
V  r
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8-40
Example 8.5 – Angular Speed
Given: a car is translating along at a speed of 55 mph
the radius of the wheel is 12.5 in.
Find: the rotational speed of the car wheel
Solution:
 miles  1 h  5280 ft 
 55



V 
h  3600 s  1 mile 
 
r
12.5 in.  1 ft 
 12 in. 
 2 rad  60 s
  77.4 rad/s 

1
revolution

 1 minute
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 77.4 rad/s

  739 rpm

8-41
Angular Acceleration
Measures the rate of change of angular velocity
change in angular speed
angular accelerati on 
time
Units : rad/s 2
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8-42
Example 8.6 – Angular Acceleration
Given: it takes 5 s for a shaft of a motor to go from zero
to 1600 rpm; assume constant angular acceleration
Find: the value of the angular acceleration of the shaft
Solution:
First, convert the angular speed from rpm to rad/s
rad
 revolution s  2 radians  1 minute 
1600

167.5



s
 minutes  1 revolution  60 seconds 
Next, calculate the angular acceleration,
change in angular speed
time
167.5 - 0 rad/s  33.5 rad

5s
s2
angular accelerati on 
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8-43
Summary
• You should have a good grasp of fundamental
dimension time and its role in engineering
analysis.
• You should recognize the role of time in
calculating
 Speed
 Acceleration
 Flow of traffic
 Flow of materials and substances
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8-44
Summary (continued)
• You should know what we mean by frequency
and period.
• You should be able to give examples of
mechanical and electrical systems with
frequency and periods.
• You should know how to define
 average and instantaneous velocity
 average and instantaneous acceleration
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8-45
Summary (continued)
• You should have a comfortable grasp of
rotational motion
 you should know the difference between
linear and rotational motion
 you should know how to define angular
velocity and acceleration
• You should understand the significance of
volume flow rate in our everyday lives and
engineering applications.
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8-46