Transcript sy16_oct26_f11a

Goals:
Lecture 15
• Chapter 11 (Work)
 Employ conservative and non-conservative forces
 Relate force to potential energy
 Use the concept of power (i.e., energy per time)
Assignment:
 HW7 due Tuesday, Nov. 1st
 For Monday: Read Chapter 12, (skip angular
momentum and explicit integration for center of mass,
rotational inertia, etc.)
Exam 2
7:15 PM Thursday, Nov. 3th
Physics 207: Lecture 15, Pg 1
Definition of Work, The basics
Ingredients: Force ( F ), displacement (  r )
Work, W, of a constant force F
acts through a displacement  r :
W = F · r
F

r
(Work is a scalar)
Work tells you something about what happened on the path!
Did something do work on you?
Did you do work on something?
Physics 207: Lecture 15, Pg 2
Net Work: 1-D Example
(constant force)
 A force F = 10 N pushes a box across a frictionless floor
for a distance x = 5 m.
F
Finish
Start
 = 0°
x
 Net Work is F x = 10 x 5 N m = 50 J
 1 N m ≡ 1 Joule (energy)
 Work reflects energy transfer
Physics 207: Lecture 15, Pg 3
Net Work: 1-D 2nd Example
(constant force)
 A force F = 10 N is opposite the motion of a box across a
frictionless floor for a distance x = 5 m.
Finish
Start
F
 = 180°
x
 Net Work is F x = -10 x 5 N m = -50 J
 Work again reflects energy transfer
Physics 207: Lecture 15, Pg 4
Work: “2-D” Example
(constant force)
 An angled force, F = 10 N, pushes a box across a
frictionless floor for a distance x = 5 m and y = 0 m
F
Finish
Start
 = -45°
Fx
x
 (Net)
Work is Fx x = F cos(-45°) x = 50 x 0.71 Nm = 35 J
 Work reflects energy transfer
Physics 207: Lecture 15, Pg 5
Exercise
Work in the presence of friction and non-contact forces
 A box is pulled up a rough (m > 0) incline by a rope-pulley-
weight arrangement as shown below.
 How many forces (including non-contact ones) are
doing work on the box ?
 Of these which are positive and which are negative?
 State the system (here, just the box)
 Use a Free Body Diagram
 Compare force and path
v
A. 2
B. 3
C. 4
D. 5
Physics 207: Lecture 15, Pg 6
Work and Varying Forces (1D)
Area = Fx x
F is increasing
Here W = F · r
becomes dW = F dx
 Consider a varying force F(x)
Fx
x
x
xf
W
  F ( x ) dx
xi
Physics 207: Lecture 15, Pg 7
Example: Work Kinetic-Energy Theorem with variable force
•
How much will the spring compress (i.e. x = xf - xi) to bring
the box to a stop (i.e., v = 0 ) if the object is moving initially at a
constant velocity (vo) on frictionless surface as shown below ?
x
Wbox  F ( x ) dx
to vo
x
x
m

f
i
   kx dx
f
Wbox
spring at an equilibrium position
xi
x
V=0
t
F
m
spring compressed
1
2
k x  mv
2
1
2
2
0
Physics 207: Lecture 15, Pg 8
Compare work with changes in potential energy
 Consider the ball moving up to height h
(from time 1 to time 2)
 How does this relate to the potential energy?
Work done by the Earth’s gravity on
the ball)
W = F  x = (-mg) -h = mgh
mg
h
U = Uf – Ui = mg 0 - mg h = -mg h
U = -W
mg
Physics 207: Lecture 15, Pg 9
Conservative Forces & Potential Energy
 If a conservative force F we can define a potential energy
function U:
W =
 F ·dr = - U
The work done by a conservative force is equal and opposite to
the change in the potential energy function. r
U
f
 or
U = Uf - Ui = - W = -
f
rf
rF • dr
i
ri
Ui
Physics 207: Lecture 15, Pg 10
A Non-Conservative Force, Friction
 Looking down on an air-hockey table with no air
flowing (m > 0).
 Now compare two paths in which the puck starts
out with the same speed (Ki path 1 = Ki path 2) .
Path 2
Path 1
Physics 207: Lecture 15, Pg 11
A Non-Conservative Force
Path 2
Path 1
Since path2 distance >path1 distance the puck will be
traveling slower at the end of path 2.
Work done by a non-conservative force irreversibly
removes energy out of the “system”.
Here WNC = Efinal - Einitial < 0  and reflects Ethermal
Physics 207: Lecture 15, Pg 12
A child slides down a playground slide at
constant speed. The energy transformation
is
A.
B.
C.
D.
E.
UK
U  ETh
KU
K ETh
There is no transformation because
energy is conserved.
Physics 207: Lecture 15, Pg 13
Conservative Forces and Potential Energy
 So we can also describe work and changes in
potential energy (for conservative forces)
U = - W
 Recalling (if 1D)
W = Fx x
 Combining these two,
U = - Fx x
 Letting small quantities go to infinitesimals,
dU = - Fx dx
 Or,
Fx = -dU / dx
Physics 207: Lecture 15, Pg 14
Equilibrium
 Example
 Spring: Fx = 0 => dU / dx = 0 for x=xeq
The spring is in equilibrium position
 In general: dU / dx = 0  for ANY function
establishes equilibrium
U
stable equilibrium
U
unstable equilibrium
Physics 207: Lecture 15, Pg 15
Work & Power:
 Two cars go up a hill, a Corvette and a ordinary Chevy




Malibu. Both cars have the same mass.
Assuming identical friction, both engines do the same amount
of work to get up the hill.
Are the cars essentially the same ?
NO. The Corvette can get up the hill quicker
It has a more powerful engine.
Physics 207: Lecture 15, Pg 16
Work & Power:
 Power is the rate, J/s, at which work is done.
 Average Power is,
 Instantaneous Power is,
W
P 
t
dW
P
dt
 If force constant, W= F x = F (v0 t + ½ at2)
and
P = W / t = F (v0 + at)
Physics 207: Lecture 15, Pg 17
Work & Power:
Average
Power:
W
P 
t
1 W = 1 J / 1s
Example:
 A person, mass 80.0 kg, runs up 2 floors (8.0 m). If
they climb it in 5.0 sec, what is the average power
used?
 Pavg = F h / t = mgh / t = 80.0 x 9.80 x 8.0 / 5.0 W
 P = 1250 W
Physics 207: Lecture 15, Pg 18
Exercise
Work & Power
 Starting from rest, a car drives up a hill at constant
acceleration and then suddenly stops at the top.
 The instantaneous power delivered by the engine during this
drive looks like which of the following,
A. Top
time
B. Middle
C. Bottom
time
time
Physics 207: Lecture 15, Pg 19
Chap. 12: Rotational Dynamics
 Up until now rotation has been only in terms of circular motion
with ac = v2 / R and | aT | = d| v | / dt
 Rotation is common in the world around us.
 Many ideas developed for translational motion are transferable.
Physics 207: Lecture 15, Pg 20
Conservation of angular momentum has consequences
Katrina
How does one describe rotation (magnitude and direction)?
Physics 207: Lecture 15, Pg 21
Rotational Variables
 Rotation about a fixed axis:
 Consider a disk rotating about
an axis through its center:


 Recall :
d 2


(rad/s)  vTangential /R
dt T
(Analogous to the linear case v 
dx )
dt
Physics 207: Lecture 15, Pg 22
System of Particles (Distributed Mass):
 Until now, we have considered the behavior of very simple
systems (one or two masses).
 But real objects have distributed mass !
 For example, consider a simple rotating disk and 2 equal
mass m plugs at distances r and 2r.

1
2
 Compare the velocities and kinetic energies at these two
points.
Physics 207: Lecture 15, Pg 23
For these two particles
1 K= ½ m v2

2 K= ½ m (2v)2 = 4 ½ m v2
 Twice the radius, four times the kinetic energy
K  mv  mv  m(r1 )  m(r2 )
1
2
2
1
2
1
4
2
2
1
2
K  [mr1  mr2 ]
1
2
2
1
2
2
1
2
2
Physics 207: Lecture 15, Pg 24
2
For these two particles
1 K= ½ m v2

2 K= ½ m (2v)2 = 4 ½ m v2
K  [mr1  mr2 ]
1
2
2
K  [  mr ]
1
2
2
1
2
2
2
2
All
mass
points
Physics 207: Lecture 15, Pg 25
For these two particles

K  [  mr ]
1
2
2
2
All
mass
points
K Rot  I (where I is the moment of inertia)
1
2
2
Physics 207: Lecture 15, Pg 26
Calculating Moment of Inertia
N
I   mi ri
2
where r is the distance from
the mass to the axis of rotation.
i 1
Example: Calculate the moment of inertia of four point masses
(m) on the corners of a square whose sides have length L,
about a perpendicular axis through the center of the square:
m
m
m
m
L
Physics 207: Lecture 15, Pg 27
Calculating Moment of Inertia...
 For a single object, I depends on the rotation axis!
 Example: I1 = 4 m R2 = 4 m (21/2 L / 2)2
I1 = 2mL2
m
m
m
m
I2 = mL2
I = 2mL2
L
Physics 207: Lecture 15, Pg 28
Lecture 16
Assignment:
 HW7 due Tuesday Nov. 1st
Physics 207: Lecture 15, Pg 29