Physics 218
Lecture 11
Dr. David Toback
Physics 218, Lecture XI
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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
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Chapters 7, 8 & 9 Cont
Last time:
• Work
This time
• More on Work
• Work and Energy
– Work using the Work-Energy
relationship
• Potential Energy
• Conservation of Mechanical Energy
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Work in Two Dimensions
You 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 Q. The floor
is rough and has coefficient of friction m.
Determine:
• The work done by each force
• The net work on the crate
Q
X
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What if the Force is
changing direction?
What if the Force is
changing magnitude?
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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!
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Find the work: Calculus
To 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


 F  dx
a
Fancy sum
notationIntegral
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Use an Integral for a Constant Force
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?
W 

d
0

d

F  dx   Fdx  Fx|xx0d  Fd  F 0  Fd
0
Region of integration
W=Fd
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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


F  kx
Some constant
Displacement
• Force is NOT
CONSTANT over
a
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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
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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
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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 object
Can use the change in energy to
calculate the work
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Summary of equations
Kinetic Energy =
2
½mV
W= DKE
Can use change in speed
to calculate the work, or
the work to calculate the
speed
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Multiple ways to
calculate the work
done
Multiple ways to
calculate the velocity
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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
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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= DKE = KEf-KEi
= ½mVf2 -½mVi2
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Quick Problem
I can do work on an object
and it doesn’t change the
kinetic energy.
How? Example?
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Problem Solving
How do you solve Work
and Energy problems?
BEFORE and AFTER
Diagrams
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Problem Solving
Before and After diagrams
1.What’s going on before
work is done
2.What’s going on after
work is done
Look at the energy before and
the energy after
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Before…
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After…
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Compressing a Spring
A horizontal spring has spring
constant k
1.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?
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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
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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
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Mechanical Energy
• We define the total
mechanical energy in a
system to be the kinetic
energy plus the potential
energy
• Define E≡K+U
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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 Energy
E2=E1
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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 =
2
½mv
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Problem Solving
For Conservation of Energy problems:
BEFORE and AFTER
diagrams
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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?
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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
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Compressing a Spring
A horizontal spring has spring
constant k
1.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 m?
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Roller Coaster
A 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
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Kinetic Energy
Take 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= mad
ad = ½V2
mad = ½ mV2
W = mad = ½ mV2
Kinetic Energy = ½ mV2
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Work and Kinetic Energy
• If V0 not equal to 0 then
• V2 - V02 = 2ad
• W=F.d = mad = ½m (V2 - V02)
= ½mV2- ½mV02 = D(Kinetic Energy)
W= DKE
Net Work on an object (All forces)
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A football is thrown
A 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
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Example: Gravity
• Work by Gravity
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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?
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Water Slide
Who hits the bottom with a faster speed?
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Mechanical Energy
• Consider a Conservative System
• Wnet = DK (work done ON an object)
DUTotal = -Wnet
Combine
DK = Wnet = -DUTotal
=> DK + DU = 0
Conservation of Energy
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Conservation of Energy
• Define E=K+U
DK + DU = 0 => (K2-K1) +(U2-U1)=0
K2+U2 = K1+U1
Conservation of Mechanical Energy
E2=E1
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Conservative vs. Non-Conservative
Forces
• Nature likes to “conserve” certain types
of things
• Keep them the same
• Kinda like conservative politicians
• Conservationists
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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)
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Closed Loops
Another 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?
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Is Friction a Conservative
Force?
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Integral Examples we know…
V   a dt 
 at
0
dt
Where a is a constant
 a(
t 01
0 1
)  v0
 a(
t1
1
)  v0
V  at  v0
X   (at  v0 ) dt 
 a(
t 11
11
 (at
)  v0(
1
 v0t ) dt
t 01
0 1
0
)  x0  a(
t2
2
)  v0t  x0
X  at  v0t  x0
1
2
2
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Work done to stretch a Spring
Stretch a spring from x  0 to x  X
Work by Person W
Xf
X
Xi
0
W   F  dl   FP  dx
X
1 2X 1 2
  kx  dx  k x  kX
0
2
2
0
1 2
W  kX
2
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Robot Arm
A robot arm has a
funny Force equation in
1-dimension
 3x 
F(x)  F0 1  2 
x
0 

where F0 and X0 are
2
constants.
What is the work done
to move a block from
position X1 to position
X2?
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Stretch a Spring
A person pulls on a spring and stretches it a
from the equilibrium point for a total
distance D. At this distance the force
required to keep the spring stretched is F.
How much work is done on the spring in terms
of the given variables?
Can we use F.d?
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