Physics 106P: Lecture 12 Notes

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Average 78.5
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Physics 211 Lecture 11Todays Agenda

• Conservative forces potential energy - review
• Conservation of total mechanical energy
• Example pendulum
• Non-conservative forces
• friction
• General work/energy theorem
• Example problem

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Conservative Forces

• We have seen that the work done by gravity does
  not depend on the path taken.

m
R2
R1
M
m
h
Wg -mgh
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Lecture 11, Act 1Work 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 earths surface and has kinetic
  energy K2 when it hits. RE is the radius of the
  earth.
• What is K2 / K1?

2RE
RE
RE
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Lecture 11, Act 1Solution

• Since energy is conserved, DK WG.

2RE
RE
RE
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Lecture 11, Act 1 Solution

• For the first rock

RE
RE
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Conservative Forces

• We have seen that the work done by a conservative
  force does not depend on the path taken.

W2
W1 W2
W1

• Therefore the work done in a closed path is 0.

W2
WNET W1 - W2 W1 - W1 0
W1
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Lecture 11, Act 2Conservative Forces

• The pictures below show force vectors at
  different points in space for two forces. Which
  one is conservative ?

(a) 1 (b) 2 (c)
both
y
y
(1)
(2)
x
x
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Lecture 11, Act 2Solution

• Consider the work done by force when moving along
  different paths in each case

WA WB
WA gt WB
(1)
(2)
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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 gt 0!
• Free kinetic energy!!

WNET 10 J DK
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Potential Energy Recap

• For any conservative force we can define a
  potential energy function U such that
• The potential energy function U is always defined
  onlyup to an additive constant.
• You can choose the location where U 0 to be
  anywhere convenient.

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Conservative Forces Potential Energies (stuff
you should know)
Work W(1-2)
Change in P.E ?U U2 - U1
P.E. function U
Force F

Fg -mg j
-mg(y2-y1)
mg(y2-y1)
mgy C

Fg r
Fs -kx
(R is the center-to-center distance, x is the
spring stretch)
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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.
• E K U is constant!!!
• Both K and U can change, but E K U remains
  constant.
• But well see that if non-conservative forces act
  then energy can be dissipated into other modes
  (thermal,sound)

E K U ?E ?K ?U W ?U W
(-W) 0

• using ?K W
• using ?U -W

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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
  wheredoes this happen?
• To what height h2 does it rise on the other side?

m
h1
h2
v
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Example The simple pendulum

• Kineticpotential energy is conserved since
  gravity is a conservative force (E K U is
  constant)
• Choose y 0 at the bottom of the swing, and U
  0 at y 0 (arbitrary choice)E 1/2mv2 mgy

y
h1
h2
y 0
v
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Example The simple pendulum

• E 1/2mv2 mgy.
• Initially, y h1 and v 0, so E mgh1.
• Since E mgh1 initially, E mgh1 always since
  energy is conserved.

y
y 0
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Example The simple pendulum

• 1/2mv2 will be maximum at the bottom of the
  swing.
• So at y 0 1/2mv2 mgh1 v2
  2gh1

y
y h1
h1
y 0
v
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Example The simple pendulum

• Since E mgh1 1/2mv2 mgy it is clear that
  the maximum height on the other side will be at y
  h1 h2 and v 0.
• The ball returns to its original height.

y
y h1 h2
y 0
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Example The simple pendulum
Bowling

• The ball will oscillate back and forth. The
  limits on its height and speed are a consequence
  of the sharing of energy between K and U. E
  1/2mv2 mgy K U constant.

y
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Example The simple pendulum

• We can also solve this by choosing y 0 to be at
  the original position of the mass, and U 0 at y
  0.E 1/2mv2 mgy.

y
y 0
h1
h2
v
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Example The simple pendulum

• E 1/2mv2 mgy.
• Initially, y 0 and v 0, so E 0.
• Since E 0 initially, E 0 always since energy
  is conserved.

y
y 0
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Example The simple pendulum

• 1/2mv2 will be maximum at the bottom of the
  swing.
• So at y -h1 1/2mv2 mgh1
  v2 2gh1

y
Same as before!
y 0
h1
y -h1
v
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Example The simple pendulum
Galileos Pendulum

• Since 1/2mv2 - mgh 0 it is clear that the
  maximum height on the other side will be at y 0
  and v 0.
• The ball returns to its original height.

y
y 0
Same as before!
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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
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Example Airtrack Glider
Glider

• Kineticpotential energy is conserved since all
  forces are conservative.
• Choose initial configuration to have U0.?K
  -?U

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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
h
v1
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Problem Hotwheel...

• KU energy is conserved, so ?E 0 ?K -
  ?U
• Moving down a distance d, ?U -mgd, ?K
  1/2mv12
• Solving for the speed

d
h
v1
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Problem Hotwheel...

• At the end, we are a distance d - h below our
  starting point.
• ?U -mg(d - h), ?K 1/2mv22
• Solving for the speed

v2
d - h
d
h
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Lecture 11, Act 3Potential 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)
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Lecture 11, Act 3Solution

• The displacement of (1) from equilibrium will be
  half of that of (2) (each spring exerts half of
  the force needed to balance mg)

0
d
2d
(1)
(2)
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Lecture 11, Act 3Solution
0
d
2d
(1)
(2)
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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!

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Energy dissipation e.g. sliding friction

• As the parts scrape by each otherthey start
  small-scale vibrations,which transfer kinetic
  and potentialenergy into atomic motions

The atoms vibrations go back and forth- they
have energy, but no average momentum.
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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
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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 gt Wpath 1

B
path 1
path 2
A
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Generalized Work/Energy Theorem

• Suppose FNET FC FNC (sum of conservative and
  non-conservative forces).
• The total work done is WNET WC WNC
• The Work/Kinetic Energy theorem says that WNET
  ?K.
• WNET WC WNC ?K
  
• WNC ?K - WC
• But WC -?U
• So WNC ?K ?U ?Emechanical

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Generalized Work/Energy Theorem
WNC ?K ?U ?Emechanical

• The change in kineticpotential energy of a
  system is equal to the work done on it by
  non-conservative forces. Emechanical KU of
  system not conserved!
• If all the forces are conservative, we know that
  KU energy is conserved ?K ?U ?Emechanical
  0 which says that WNC 0, which makes sense.
• If some non-conservative force (like friction, a
  push or a pull) does work, KU energy will
  not be conserved and WNC ?E, which also makes
  sense.

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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
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Problem Block Sliding with Friction...

• Using WNC ?K ?U
• As before, ?U -mgd
• WNC work done by friction -?kmgx.
• ?K 0 since the block starts out and ends up at
  rest.
• WNC ?U -?kmgx -mgd x d / ?k

d
?k
x
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Recap of todays lecture

• Conservative forces potential energy - review
• Conservation of Total Mechanical Energy
• Examples pendulum, airtrack, Hotwheel car
• Non-conservative forces
• friction
• General work/energy theorem
• Example problem