Lecture 4 Mechanizationsites.fas.harvard.edu/~scia52/Lectures/Lecture04.pdf · Spring 2006 ©...

Spring 2006 © Science A 52 FHA + MBM1

Lecture 4February 13, 2006

• Mechanization

•US Energy Consumptionfrom all forms of energy from

1850 - 2000


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”


The Questions is: how toCalculate 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 casethe lever.

• Simplify the situation - keep only theessential feature.


The Lever - continuedbegin drawing a free body or force diagram


complete force diagram


Now Apply Newton’s Laws

• Sum of all Forces and Moments mustbe zero for the body to be at rest.

Forces = 0all

!

Momentsall

! = 0


Looking first at the forces!

• Forces are vectors and we must write vectorequations

• All forces are in the vertical direction• Therefore,

[B - W - P] ( j ) =0 Important Resultor,

B - W - P = 0


What next?

• We have just one equation and twounknowns. We know the load W butnot the B the force of the fulcrum on thelever nor the force F needed to lift theload. What to do? We need anotherequation, and the Moment equation isthere to be used.


The moment equation

• We can take moments of the forces about theorigin of the given coordinate system system.This eliminates the force B from the momentequation since the force passes through theorigin. 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 withoutusing the force equation, but if we wished tocalculate the force of the fulcrum on the lever wewould have needed to use the force equation.


Lever continued• In the course notes on Mechanization I have

gone through the calculation to show that thework done by the force P pushing down onthe lever (lifting the weight W) is the same asthe work done on W lifting it against the pullof gravity. You do positive work in pushingdown and negative work is done lifting theweight against gravity.

• You do work on the system by pushing down,and the work is used to increase the potentialenergy of the weight. Go over these conceptsin section if you have problems with theseideas.


levers are everywhere


another lever


how to analyze the pliers


The inclined plane

The basic geometry of the inclined plan are shown in thesketch. This is not quite a diagram of all of the forcesacting on the weight W. The force of the incline pushingon the weight has not been drawn. Without friction thereis no component of this force along the direction of theincline.


Force balance along the direction of the incline

Forces = 0 = F !W sin"i

#

• We assumed that the coal train was operating at aconstant speed; that is, there was no accelerationIn the direction of the incline. Hence the sum of the forces along the direction of the incline =0.


incline plane - continued

F • v = rate of doing work = Fv

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 notesyou - the reader- are asked to calculate the speed.


inclined plane - continued

• The rate of doing work or power is simply the dotproduct of the force and the velocity of the mass -the coal train.

• In the notes the hp of the engines is expressed inunits of the rate of doing work. The engine rate ofdoing work is just the force of the pull or push of theengines dotted into train’s velocity. The force andvelocity are in the same direction so the dot productis just th magnitude of the product of the force andthe speed. The train does move quite slowing, justas the New Yorker reported.


inclined plane - continued

• The notes calculate the work done inmoving the weight a distance S alongthe incline and show that it is exactlythe same at the work done in lifting theweight 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.


Power transmitter from onepulley to another


connected pulleys

Here is sketch of the essential of the spinning wheel withoutthe force shown that are necessary to hold the shafts of the “pulleys” fixed in place.


connected pulleys

• The speed of any element of a rotatingpulley is simply the product of theangular speed times the distance fromthe axle; that is .

• Or for the two pulleys we have for thespeed of the connecting belts.

!1a =!

2b

! r


connected pulleys

•Looking first at the bigger pulley, here is a sketch showingthe belt tensions and the corresponding forces on the sleeve bearings that keep the pulley inplace.


connected pulleys

T1! T

2" b( ) "#1

" a / b = P3$ P

2$ P

1

•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 is the power. But we also know that !

2!1" a / b =!

2

•Therefore on can make the substitution and the results below follow.

The conclusion is that in the absence of frictional losses,all of the power put into the system is transmitted throughall of the pulleys connected by belts.


gears

• I will leave you to read the section on gears inthe notes. If there are questions left after youread the material I will try and answer themthe next time I lecture. Gear systems aresimilar to pulley systems. The advantage isthat metal gears can transmit more powerthan belt without slipping. Belt systems dohave advantages - note the belt drivesconnecting generators to the car engine.


Now for US Energy Sources


U.S. Energy Use in 2000

• Our current primary sources ofenergy.

• How have the sources variedover time.


Conversions of units of energy


1 BTU is 1,055 JOULES• So a Exajoule plot is similar tothe Quad Btu plot - the numbersare just 1,055 times larger in the

Exajoule plot.

• A Btu is 1,055 times larger than ajoule.


What were our sources ofenergy in the past?


What were our sources ofenergy in the past?

• At the beginning of the industrialrevolution in the US the major energysource was wood

• From 1800 until 1850 wood was burnedand work animals did work in the fieldsand pulled the wagons.


The next series of charts showthe U.S. sources of energy

from1850 to 2000

But first the data sources


Data references for the chartsof 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 NewSurvey, (The Twentieth Century Fund, NewYork, 1955), pp. 1114.

• Data for years 1950 to 2000 called series 2on the charts see UCRL-ID-129990-00,U.S.Energy Flow -2000,Gina V. Kaiper, Feb. 2002


Comments for EnergySources 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

Lecture 4 Mechanizationsites.fas.harvard.edu/~scia52/Lectures/Lecture04.pdf · Spring 2006 ©...

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