CHAPTER 8 Time and Time-Related Parameters © 2011 Cengage Learning Engineering. All Rights Reserved. 8-1 Fundamental Dimensions • Length • Time and time-related parameters • Mass • Temperature • Electric current • Amount of substance • Luminous intensity © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-3 Outline (continued) • Learn about engineering variables involving length & time linear velocity linear acceleration volume flow rate rotational motion © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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? © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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? © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-11 Standard Time Zones © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-16 Examples of Frequencies of Various Electrical and Electronic Systems © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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) © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-22 Time-Related Variables – Linear Velocities • Provide a measure of how fast an object moves • Examples? © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-24 Examples of Some Speeds © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-26 Acceleration Can we have an acceleration without a change in speed? © 2011 Cengage Learning Engineering. All Rights Reserved. 8-27 Acceleration Due to Gravity © 2011 Cengage Learning Engineering. All Rights Reserved. 8-28 Speed, Acceleration, and Distance speed, acceleration, and distance traveled by a falling object, neglecting the air resistance © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 8-38 Angular Motion Examples • Shafts • Wheels • Gears • Drills • Fan or pump impellers • DVD or hard drives • Helicopter blades © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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 © 2011 Cengage Learning Engineering. All Rights Reserved. 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. © 2011 Cengage Learning Engineering. All Rights Reserved. 8-46