Download Lecture 11

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Mathematical physics wikipedia , lookup

Renormalization group wikipedia , lookup

Eigenstate thermalization hypothesis wikipedia , lookup

Energy profile (chemistry) wikipedia , lookup

Molecular dynamics wikipedia , lookup

Internal energy wikipedia , lookup

Transcript
Physics 218
Lecture 11
Dr. David Toback
Physics 218, Lecture XI
1
Chapters 6 & 7 Seven Cont
Last time:
• Work
This time
• Work and Energy
– Work using the Work-Energy
relationship
• Potential Energy
• Conservation of Mechanical Energy
Physics 218, Lecture XI
2
Physics 218, Lecture XI
3
Physics 218, Lecture XI
4
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
2
Kinetic Energy = ½mV
Physics 218, Lecture XI
5
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
Physics 218, Lecture XI
6
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
Physics 218, Lecture XI
7
Multiple ways to
calculate the work
done
Multiple ways to
calculate the velocity
Physics 218, Lecture XI
8
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
9
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
Physics 218, Lecture XI
10
Quick Problem
I can do work on an object
and it doesn’t change the
kinetic energy.
How? Example?
Physics 218, Lecture XI
11
Problem Solving
How do you solve Work
and Energy problems?
BEFORE and AFTER
Diagrams
Physics 218, Lecture XI
12
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
Physics 218, Lecture XI
13
Before…
Physics 218, Lecture XI
14
After…
Physics 218, Lecture XI
15
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?
Physics 218, Lecture XI
16
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
17
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
18
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
19
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
Physics 218, Lecture XI
20
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
Physics 218, Lecture XI
+ mgy
21
Problem Solving
For Conservation of Energy
problems:
BEFORE and AFTER
diagrams
Physics 218, Lecture XI
22
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
23
Next Week
• Reading for Next Time:
–Finish Chapters 6 and 7 if you
haven’t already
–Non-conservative forces &
Energy
• Homework 5 Due Monday
• Start working on HW6
Physics 218, Lecture XI
24
Physics 218, Lecture XI
25
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?
Physics 218, Lecture XI
26
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
Physics 218, Lecture XI
27
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
Physics 218, Lecture XI
28
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)
Physics 218, Lecture XI
29
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
Physics 218, Lecture XI
30
Example: Gravity
• Work by Gravity
Physics 218, Lecture XI
31
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
32
Water Slide
Who hits the bottom with a faster speed?
Physics 218, Lecture XI
33
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
Physics 218, Lecture XI
34
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
Physics 218, Lecture XI
35
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
36
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
37
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?
Physics 218, Lecture XI
38
Is Friction a Conservative Force?
Physics 218, Lecture XI
39
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
Physics 218, Lecture XI
40
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
Physics 218, Lecture XI
41
Robot Arm
A robot arm has a
funny Force equation
in 1-dimension
 3x
F(x)  F0 1  2
x0

2



where F0 and X0 are
constants.
What is the work done
to move a block from
position X1 to position
X2?
Physics 218, Lecture XI
42
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?
Physics 218, Lecture XI
43