Physics 218, Lecture XI 2
Checklist for Today•Things that were due Monday:
– Chaps. 3 and 4 HW on WebCT– Progress on 5&6 problems
•Things that were due Tuesday:– Reading for Chapter 7
•Things due for Wednesday’s Recitation:– Problems from Chap 5&6
•Things due for Today:– Read Chapters 7, 8 & 9
•Things due Monday– Chap 5&6 turned in on WebCT
Physics 218, Lecture XI 3
Last time:• WorkThis time• More on Work• Work and Energy
– Work using the Work-Energy relationship
• Potential Energy• Conservation of Mechanical Energy
Chapters 7, 8 & 9 Cont
Physics 218, Lecture XI 6
Work in Two DimensionsYou pull a crate of mass M a distance X along a horizontal floor with a constant force. Your pull has magnitude FP, and acts at an angle of . The floor is rough and has coefficient of friction .
Determine:•The work done by each force•The net work on the crate
X
Physics 218, Lecture XI 7
What if the Force is changing direction?What if the Force is
changing magnitude?
Physics 218, Lecture XI 8
What if the force or direction isn’t constant?
I exert a force over a distance for awhile, then exert a different force over a different distance (or direction) for awhile. Do this a number of times. How much work did I do?
Need to add up all the little
pieces of work!
Physics 218, Lecture XI 9
Fancy sum notationIntegr
al
Find the work: CalculusTo find the total work, we must sum up all the little pieces of work (i.e., F.d). If the force is continually changing, then we have to take smaller and smaller lengths to add. In the limit, this sum becomes an integral.
b
a
xdF
Physics 218, Lecture XI 10
Use an Integral for a Constant Force
Fd0FFd| FxFdx xdFW dx0x
d
0
d
0
Assume a constant Force, F, doing work in the same direction, starting at x=0 and continuing for a distance d. What is the work?
Region of integrationW=Fd
Physics 218, Lecture XI 11
Non-Constant Force: Springs
•Springs are a good example of the types of problems we come back to over and over again!
•Hooke’s Law
•Force is NOT CONSTANT over a distance
Some constantDisplacement
xkF
Physics 218, Lecture XI 12
Work done to stretch a Spring
How much work do you do to stretch a spring (spring constant k), at constant velocity, from x=0 to x=D?
D
Physics 218, Lecture XI 13
Kinetic Energy and Work-Energy
• Energy is another big concept in physics
• If I do work, I’ve expended energy
– It takes energy to do work (I get tired)
• If net work is done on a stationary box it speeds up. It now has energy
• We say this box has “kinetic” energy! Think of it as Mechanical Energy or the Energy of Motion
Kinetic Energy = ½mV2
Physics 218, Lecture XI 14
Work-Energy Relationship
•If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes)
•Equivalent statement: If there is a change in kinetic energy then there has been net work on an objectCan use the change in energy
to calculate the work
Physics 218, Lecture XI 15
Summary of equations
Kinetic Energy = ½mV2
W= KECan use change in speed to calculate the work, or
the work to calculate the speed
Physics 218, Lecture XI 16
Multiple ways to calculate the work
doneMultiple ways to
calculate the velocity
Physics 218, Lecture XI 17
Multiple ways to calculate work
1. If the force and direction is constant–F.d
2. If the force isn’t constant, or the angles change– Integrate
3. If we don’t know much about the forces–Use the change in kinetic energy
Physics 218, Lecture XI 18
Multiple ways to calculate velocity
If we know the forces:•If the force is constant
–F=ma →V=V0+at, or V2-V02 = 2ad
•If the force isn’t constant
–Integrate the work, and look at the change in kinetic energy W= KE = KEf-KEi
= ½mVf2 -½mVi
2
Physics 218, Lecture XI 19
Quick Problem
I can do work on an object and it doesn’t change the kinetic energy.
How? Example?
Physics 218, Lecture XI 20
Problem Solving
How do you solve Work and Energy
problems? BEFORE and AFTER
Diagrams
Physics 218, Lecture XI 21
Problem Solving
Before and After diagrams1.What’s going on
before work is done 2.What’s going on after
work is doneLook at the energy before and the energy after
Physics 218, Lecture XI 24
Compressing a SpringA horizontal spring has
spring constant k1.How much work must you
do to compress it from its uncompressed length (x=0) to a distance x=-D with no acceleration?
2.You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?
Physics 218, Lecture XI 25
Potential Energy
• Things with potential: COULD do work–“This woman has great potential as an engineer!”
• Here we kinda mean the same thing
• E.g. Gravitation potential energy:
–If you lift up a brick it has the potential to do damage
Physics 218, Lecture XI 26
Example: Gravity & Potential Energy
You lift up a brick (at rest) from the ground and then hold it at a height Z
•How much work has been done on the brick?
•How much work did you do?•If you let it go, how much work will be done by gravity by the time it hits the ground?
We say it has potential energy: U=mgZ
–Gravitational potential energy
Physics 218, Lecture XI 27
Mechanical Energy
•We define the total mechanical energy in a system to be the kinetic energy plus the potential energy
•Define E≡K+U
Physics 218, Lecture XI 28
Conservation of Mechanical Energy
• For some types of problems, Mechanical Energy is conserved (more on this next week)
• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick
K2+U2 = K1+U1
Conservation of Mechanical EnergyE2=E1
Physics 218, Lecture XI 29
Problem Solving• What are the types of examples
we’ll encounter?– Gravity– Things falling– Springs
• Converting their potential energy into kinetic energy and back again
E = K + U = ½mv2 + mgy
Physics 218, Lecture XI 30
Problem Solving
For Conservation of Energy problems:
BEFORE and AFTER diagrams
Physics 218, Lecture XI 31
Quick Problem
We drop a ball from a height D above the ground
Using Conservation of Energy, what is the speed just before it hits the ground?
Physics 218, Lecture XI 32
Next Week• Reading for Next Time:
–Finish Chapters 7, 8 and 9 if you haven’t already
–Non-conservative forces & Energy
• Chapter 5&6 Due Monday on WebCT
• Start working on Chapter 7 for recitation next week
Physics 218, Lecture XI 34
Compressing a SpringA horizontal spring has spring
constant k1.How much work must you do
to compress it from its uncompressed length (x=0) to a distance x= -D with no acceleration?
2.You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?
3.What is the speed if there is friction with coefficient ?
Physics 218, Lecture XI 35
Roller CoasterA Roller Coaster of
mass M=1000kg starts at point A.
We set Y(A)=0. What is the potential energy at height A, U(A)?
What about at B and C?
What is the change in potential energy as we go from B to C?
If we set Y(C)=0, then what is the potential energy at A, B and C? Change from B to C
Physics 218, Lecture XI 36
Kinetic EnergyTake a body at rest, with mass m, accelerate
for a while (say with constant force over a distance d). Do W=Fd=mad:
• V2- V02 = 2ad= V2
ad = ½V2
• W = F.d = (ma) .d= madad = ½V2
mad = ½ mV2
W = mad = ½ mV2
Kinetic Energy = ½ mV2
Physics 218, Lecture XI 37
Work and Kinetic Energy
• If V0 not equal to 0 then
• V2 - V02 = 2ad
• W=F.d = mad = ½m (V2 - V02)
= ½mV2- ½mV02 = (Kinetic Energy)
W= KE
Net Work on an object (All forces)
Physics 218, Lecture XI 38
A football is thrownA 145g football starts at rest and is
thrown with a speed of 25m/s.
1. What is the final kinetic energy?2. How much work was done to reach
this velocity?
We don’t know the forces exerted by the arm as a function of time, but this allows us to sum them all up to calculate the work
Physics 218, Lecture XI 40
Potential Energy in General
• Is the potential energy always equal to the work done on the object?– No, non-conservative forces– Other cases?
• What about for conservative forces?
Physics 218, Lecture XI 42
Mechanical Energy
• Consider a Conservative System
• Wnet = K (work done ON an object)
UTotal = -Wnet Combine
K = Wnet = -UTotal
=> K + U = 0 Conservation of Energy
Physics 218, Lecture XI 43
Conservation of Energy
• Define E=K+U
K + U = 0 => (K2-K1) +(U2-U1)=0
K2+U2 = K1+U1
Conservation of Mechanical EnergyE2=E1
Physics 218, Lecture XI 44
Conservative vs. Non-Conservative Forces
• Nature likes to “conserve” certain types of things
• Keep them the same
• Kinda like conservative politicians
• Conservationists
Physics 218, Lecture XI 45
Conservative Forces• Physics has the same meaning. Except
nature ENFORCES the conservation. It’s not optional, or to be fought for.
“A force is conservative if the work done by a force on an object moving from one point to another point depends only on the initial and final positions and is independent of the particular path taken”
• (We’ll see why we use this definition later)
Physics 218, Lecture XI 46
Closed LoopsAnother definition: A force is conservative
if the net work done by the force on an object moving around any closed path is zero
This definition and the previous one give the same answer. Why?
Physics 218, Lecture XI 48
002
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constant a is Where
xtvatX
xtv) a(x)(v) a(
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vatV
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a dt at a dt V
ttt
t
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Integral Examples we know…
Physics 218, Lecture XI 49
Work done to stretch a Spring
2
2X
0
X
0
2
X
0
P
X
X
kX2
1W
kX2
1xk
2
1dxkx
dxFdlF W
Person Wby Work X x to0 xfrom spring aStretch
f
i
Physics 218, Lecture XI 50
Robot ArmA robot arm has a funny Force equation in 1-dimension
where F0 and X0 are constants.What is the work done to move a block from position X1 to position X2?
20
2
0 x
3x1F F(x)
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