Physics 211: Lecture 11
Today’s Agenda
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



Conservative forces & potential energy - review
Conservation of “total mechanical energy”
 Example: pendulum
Non-conservative forces
 friction
General work/energy theorem
Example problem
Physics 211: Lecture 11, Pg 1
Exam 1
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Question 12:
Physics 211: Lecture 11, Pg 2
Exam 1
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Question 15:
Physics 211: Lecture 11, Pg 3
Exam 1
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Question 4:
Physics 211: Lecture 11, Pg 4
Exam 1
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Question 18:
Physics 211: Lecture 11, Pg 5
Exam 1
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Question 24:
Physics 211: Lecture 11, Pg 6
Conservative Forces:

We have seen that the work done by gravity does not
depend on the path taken.
m
 1
1 

Wg  GMm


R
R
 2
1 
R2
R1
M
m
h
Wg = -mgh
Physics 211: Lecture 11, Pg 7
Lecture 11, Act 1
Work & Energy

A rock is dropped from a distance RE above the surface of the earth,
and is observed to have kinetic energy K1 when it hits the ground.
An identical rock is dropped from twice the height (2RE) above the
earth’s surface and has kinetic energy K2 when it hits. RE is the
radius of the earth.
 What
(a)
is K2 / K1?
2
2RE
(b)
(c)
3
2
4
3
RE
RE
Physics 211: Lecture 11, Pg 8
Lecture 11, Act 1
Solution
2RE
RE
RE
Physics 211: Lecture 11, Pg 9
Lecture 11, Act 1
Solution
2RE
RE
RE
Physics 211: Lecture 11, Pg 10
Conservative Forces:

We have seen that the work done by a conservative force
does not depend on the path taken.
W2
W1 = W2

Therefore the work done in a
closed path is 0.
W1
W2
WNET = W1 - W2
= W1 - W1 = 0
W1
Physics 211: Lecture 11, Pg 11
Lecture 11, Act 2
Conservative Forces

The pictures below show force vectors at different points in
space for two forces. Which one is conservative ?
(a) 1
(b) 2
y
(c) both
y
x
(1)
x
(2)
Physics 211: Lecture 11, Pg 12
Lecture 11, Act 2
Solution

Consider the work done by force when moving along different
paths in each case:
WA = WB
(1)
WA > WB
(2)
Physics 211: Lecture 11, Pg 13
Lecture 11, Act 2

In fact, you could make money on type (2) if it ever existed:
 Work done by this force in a “round trip” is > 0!
 Free kinetic energy!!
WNET = 10 J = DK
W=0
W = 15 J
Note: NO REAL
FORCES OF THIS
TYPE EXIST, SO
FAR AS WE KNOW
W = -5 J
W=0
Physics 211: Lecture 11, Pg 14
Potential Energy Recap:

For any conservative force we can define a potential
energy function U such that:
F.dr
S2
DU = U2 - U1 = -W = -
S1

The potential energy function U is always defined only
up to an additive constant.
 You
can choose the location where U = 0 to be
anywhere convenient.
Physics 211: Lecture 11, Pg 15
Conservative Forces & Potential Energies
(stuff you should know):
Work
W(1-2)
Force
F
^
Fg = -mg j
Fg = 
GMm ^
r
2
R
Fs = -kx
-mg(y2-y1)
 1
1 

GMm
R  R 
 2
1 

1
k x22  x12 
2
Change in P.E
DU = U2 - U1
mg(y2-y1)
P.E. function
U
mgy + C
 1
GMm
1 


C
 GMm

R

R
 2 R1 
1
k x22  x12 
2
1 2
kx  C
2
(R is the center-to-center distance, x is the spring stretch)
Physics 211: Lecture 11, Pg 16
Lecture 11, Act 3
Potential Energy

All springs and masses are identical. (Gravity acts down).
 Which of the systems below has the most potential energy
stored in its spring(s), relative to the relaxed position?
(a) 1
(b) 2
(c) same
(1)
(2)
Physics 211: Lecture 11, Pg 17
Lecture 11, Act 3
Solution
(1)
(2)
Physics 211: Lecture 11, Pg 18
Lecture 11, Act 3
Solution
Physics 211: Lecture 11, Pg 19
Conservation of Energy

If only conservative forces are present, the total kinetic plus
potential energy of a system is conserved,
i.e. the
total “mechanical energy” is conserved.
 (note: E=Emechanical throughout this discussion)
E=K+U
DE = DK + DU
 using DK = W
= W + DU
= W + (-W) = 0  using DU = -W
E = K + U is constant!!!


Both K and U can change, but E = K + U remains constant.
But we’ll see that if non-conservative forces act then energy can
be dissipated into other modes (thermal,sound)
Physics 211: Lecture 11, Pg 20
Example: The simple pendulum

Suppose we release a mass m from rest a distance h1
above its lowest possible point.
 What is the maximum speed of the mass and where
does this happen?
 To what height h2 does it rise on the other side?
m
h1
h2
v
Physics 211: Lecture 11, Pg 21
Example: The simple pendulum
y
y=0
h1
h2
v
Physics 211: Lecture 11, Pg 22
Example: The simple pendulum
y
y=0
Physics 211: Lecture 11, Pg 23
Example: The simple pendulum
y
y = h1
y=0
h1
v
Physics 211: Lecture 11, Pg 24
Example: Airtrack & Glider

A glider of mass M is initially at rest on a horizontal
frictionless track. A mass m is attached to it with a
massless string hung over a massless pulley as shown.
What is the speed v of M after m has fallen a distance d ?
v
M
m
d
v
Physics 211: Lecture 11, Pg 31
Example: Airtrack & Glider
Glider
v
M
m
d
Physics 211: Lecture 11, Pg 32
Problem: Hotwheel

A toy car slides on the frictionless track shown below. It
starts at rest, drops a distance d, moves horizontally at
speed v1, rises a distance h, and ends up moving
horizontally with speed v2.
 Find v1 and v2.
v2
d
v1
h
Physics 211: Lecture 11, Pg 33
Problem: Hotwheel...
d
v1
h
Physics 211: Lecture 11, Pg 34
Problem: Hotwheel...
d-h
d
v2
h
Physics 211: Lecture 11, Pg 35
Non-conservative Forces:

If the work done does not depend on the path taken, the
force is said to be conservative.

If the work done does depend on the path taken, the force
is said to be non-conservative.

An example of a non-conservative force is friction.
 When pushing a box across the floor, the amount of
work that is done by friction depends on the path taken.
» Work done is proportional to the length of the path!
Physics 211: Lecture 11, Pg 36
Energy dissipation: e.g. sliding friction

As the parts scrape by each other
they start small-scale vibrations,
which transfer kinetic and potential
energy into atomic motions
The atoms’ vibrations go
back and forththey have energy,
but no average momentum.
Physics 211: Lecture 11, Pg 37
Non-conservative Forces: Friction

Suppose you are pushing a box across a flat floor. The mass
of the box is m and the coefficient of kinetic friction is k.

The work done in pushing it a distance D is given by:
Wf = Ff • D = -kmgD.
Ff = -kmg
D
Physics 211: Lecture 11, Pg 38
Non-conservative Forces: Friction

Since the force is constant in magnitude and opposite in direction to
the displacement, the work done in pushing the box through an
arbitrary path of length L is just
Wf = -mgL.
Clearly, the work done depends on the path taken.

Wpath 2 > Wpath 1

B
path 1
path 2
A
Physics 211: Lecture 11, Pg 39
Generalized Work/Energy Theorem:
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Suppose FNET = FC + FNC (sum of conservative and nonconservative forces).

The total work done is: WNET = WC + WNC

The Work/Kinetic Energy theorem says that: WNET = DK.
 WNET = WC + WNC = DK

WNC = DK - WC

But WC = -DU
So
WNC = DK + DU = DEmechanical
Physics 211: Lecture 11, Pg 40
Problem: Block Sliding with Friction

A block slides down a frictionless ramp. Suppose the horizontal
(bottom) portion of the track is rough, such that the coefficient of
kinetic friction between the block and the track is k.
 How far, x, does the block go along the bottom portion of the
track before stopping?
d
k
x
Physics 211: Lecture 11, Pg 42
Problem: Block Sliding with Friction...
d
k
x
Physics 211: Lecture 11, Pg 43
Recap of today’s lecture





Conservative forces & potential energy - review
Conservation of “Total Mechanical Energy”
(Text: 7-1)
 Examples: pendulum, airtrack, Hotwheel car
Non-conservative forces
(Text: 6-4)
 friction
General work/energy theorem
(Text: 7-2)
Example problem
Physics 211: Lecture 11, Pg 44