Lecture 4 February 13, Mechanization

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1 Lecture 4 February 13, 2006 Mechanization US Energy Consumption from all forms of energy from

2 The Lever Archimedes (Greek mathematician, 287 to 212 B.C.) who is believed to have said, Give me a place to stand and I will move the Earth 2

3 The Questions is: how to Calculate the Force Needed to Lift the Load? Because of Newton we now have a way to solve this force problem. First draw a diagram indicating all of the forces acting on the object - in this case the lever. Simplify the situation - keep only the essential feature. 3

4 The Lever - continued begin drawing a free body or force diagram 4

5 complete force diagram 5

6 Now Apply Newton s Laws Sum of all Forces and Moments must be zero for the body to be at rest.! all Forces = 0! Moments = 0 all 6

7 Looking first at the forces! Forces are vectors and we must write vector equations All forces are in the vertical direction Therefore, [B - W - P] ( j ) =0 Important Result or, B - W - P = 0 7

8 What next? We have just one equation and two unknowns. We know the load W but not the B the force of the fulcrum on the lever nor the force F needed to lift the load. What to do? We need another equation, and the Moment equation is there to be used. 8

9 The moment equation We can take moments of the forces about the origin of the given coordinate system system. This eliminates the force B from the moment equation since the force passes through the origin. Therefore, -d (P k) + a (W k) = 0, or, d P = a W or P= W (a/d) Note: We could have gotten this answer without using the force equation, but if we wished to calculate the force of the fulcrum on the lever we would have needed to use the force equation. 9

10 Lever continued In the course notes on Mechanization I have gone through the calculation to show that the work done by the force P pushing down on the lever (lifting the weight W) is the same as the work done on W lifting it against the pull of gravity. You do positive work in pushing down and negative work is done lifting the weight against gravity. You do work on the system by pushing down, and the work is used to increase the potential energy of the weight. Go over these concepts in section if you have problems with these ideas. 10

11 levers are everywhere 11

12 levers are everywhere 12

13 another lever 13

14 how to analyze the pliers 14

15 The inclined plane The basic geometry of the inclined plan are shown in the sketch. This is not quite a diagram of all of the forces acting on the weight W. The force of the incline pushing on the weight has not been drawn. Without friction there is no component of this force along the direction of the incline. 15

16 Force balance along the direction of the incline We assumed that the coal train was operating at a constant speed; that is, there was no acceleration In the direction of the incline. Hence the sum of the forces along the direction of the incline =0. # i Forces = 0 = F! W sin" 16

17 incline plane - continued In the class notes there is the real story of the coal train moving a 150 car coal train up a 1.5% grade and in the notes you - the reader- are asked to calculate the speed. F v = rate of doing work = Fv 17

18 inclined plane - continued The rate of doing work or power is simply the dot product of the force and the velocity of the mass - the coal train. In the notes the hp of the engines is expressed in units of the rate of doing work. The engine rate of doing work is just the force of the pull or push of the engines dotted into train s velocity. The force and velocity are in the same direction so the dot product is just th magnitude of the product of the force and the speed. The train does move quite slowing, just as the New Yorker reported. 18

19 inclined plane - continued The notes calculate the work done in moving the weight a distance S along the incline and show that it is exactly the same at the work done in lifting the weight through the vertical height S sin! From an energy point of view it is also clear that all of the work done by the force F must go into raising the weigh an appropriate distance against gravity. 19

20 Power transmitter from one pulley to another 20

21 connected pulleys Here is sketch of the essential of the spinning wheel without the force shown that are necessary to hold the shafts of the pulleys fixed in place. 21

22 connected pulleys The speed of any element of a rotating pulley is simply the product of the angular speed times the distance from the axle; that is! r. Or for the two pulleys we have for the speed of the connecting belts.! 1 a =! 2 b 22

23 connected pulleys Looking first at the bigger pulley, here is a sketch showing the belt tensions and the corresponding forces on the sleeve bearings that keep the pulley in place. 23

24 connected pulleys Looking at the smaller pulley, the moment on the shaft due to the belt tensions is given by the terms in the first bracket below. The moment times the the rotation rate! 2 is the power. But we also know that! 1 " a / b =! 2 Therefore on can make the substitution and the results below follow. ( T! T " b) " # " a / b = P $ P $ P The conclusion is that in the absence of frictional losses, all of the power put into the system is transmitted through all of the pulleys connected by belts. 24

25 gears I will leave you to read the section on gears in the notes. If there are questions left after you read the material I will try and answer them the next time I lecture. Gear systems are similar to pulley systems. The advantage is that metal gears can transmit more power than belt without slipping. Belt systems do have advantages - note the belt drives connecting generators to the car engine. 25

26 Now for US Energy Sources 26

27 U.S. Energy Use in 2000 Our current primary sources of energy. How have the sources varied over time. 27

28 28

29 Conversions of units of energy 29

30 1 BTU is 1,055 JOULES So a Exajoule plot is similar to the Quad Btu plot - the numbers are just 1,055 times larger in the Exajoule plot. A Btu is 1,055 times larger than a joule. 30

31 31

32 What were our sources of energy in the past? 32

33 What were our sources of energy in the past? At the beginning of the industrial revolution in the US the major energy source was wood From 1800 until 1850 wood was burned and work animals did work in the fields and pulled the wagons. 33

34 The next series of charts show the U.S. sources of energy from 1850 to 2000 But first the data sources 34

35 Data references for the charts of U.S. energy supply by source Data for the years 1850 to 1950 called series 1 see J. Frederic Dewhurst and Associates, America s Needs and Resources: A New Survey, (The Twentieth Century Fund, New York, 1955), pp Data for years 1950 to 2000 called series 2 on the charts see UCRL-ID ,U.S. Energy Flow -2000,Gina V. Kaiper, Feb

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37 Comments for Energy Sources Over time Note that coal did not overtake renewable energy as the primary source until about 1885 Renewable energy had peaked by 1870 Coal was our primary source of energy until 1950 after WWII 37

Chapter 09 Multiple Choice Test

Class: Date: Chapter 09 Multiple Choice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A simple machine can multiply: a. forces only. b. energy only.