Physics 2211: Lecture 18 Todays Agenda

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Physics 2211 Lecture 18Todays Agenda

• Review
• Non-conservative forces
• friction
• Work / Energy theorem
• Conservation of Energy
• Example problem

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Conservation of Mechanical Energy

• If only internal conservative forces are present,
  the total kinetic plus potential energy of an
  isolated system (no external work done on system)
  is conserved.
• E
  K U is constant!!! or DK DU
  0
• Both K and U can change, but E K U remains
  constant.

E K U ?E ?K ?U W ?U W
(-W) 0
using ?K W using ?U -W
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Problem Hotwheel

• A toy car slides on the frictionless track shown
  below. It starts at rest (point A), drops a
  distance d, moves horizontally at speed vB (point
  B), rises a distance h, and ends up moving
  horizontally with speed vC (point C).
• Find vB and vC.

A
C
vC
B
d
h
vB
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Problem Hotwheel...

• E K U , total energy of car-Earth system is
  conserved.
• So, E 1/2mv2 mgy constant
• Choosing point B to be at y 0
• EA 0 mgd
• At point B, EB 1/2mvB2 0
• Conservation of energy EA EB gt mgd 1/2mvB2
  

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Problem Hotwheel...

• At point C, EC 1/2mvC2 mgh
• Conservation of energy EA EC gt mgd 1/2mvC2
  mgh
• Solving for the speed

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Non-Conservative Forces Revisited

• 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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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

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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 -?kmgL.
• Clearly, the work done depends on the path taken.

• Wpath 2 gt Wpath 1

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External and Non-Conservative Internal Forces

• Remember, mechanical energy E ( K U) of a
  system is conserved only
• if the internal forces present are conservative
  AND
• if the system is isolated system (no external
  force is doing work done on the system).
• If external forces and/or internal
  non-conservative forces do work on the system,
  then E is not conserved. The net work, WNET, done
  on the system is

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External and Non-Conservative Internal Forces

• From the work / kinetic energy theorem,

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Example Lifting a book with your handWhat
force must be applied?

• System Book-Earth-Atmosphere (air)

• Work done by external force (hand)

• Work done by internal non-conservative force (air
  resistance)

• DE DU mgDr (DK 0)

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Example Lifting a book...
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Work / Energy Theorem

• The change in mechanical energy (kinetic
  potential) of a system is equal to the work done
  on the system by external forces and the work
  done within the system by internal
  non-conservative forces.
• Mechanical energy, E K U, of system not
  conserved!

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Conservation of Energy

• WNC done by the internal non-conservative forces
  (e.g., friction) serves as a mechanism to
  transform system energy among its various forms
  (e.g., mechanical to thermal).
• If DEINT represents the change in system
  internal energy other than mechanical, then from
  experiment WNC - DEINT and

• If WEXT 0, then

Total energy of an isolated system is conserved!
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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

• System Block, ramp, Earth
• Using WNC ?K ?U (WEXT 0 )
• As before, ?U -mgd
• WNC work done by friction -?kmgx.
• ?K 0 since the block starts out and ends up at
  rest.

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Recap of todays lecture

• Non-conservative forces
• friction
• Work / Energy theorem
• Conservation of Energy
• Example problem
• Review Chapters 6 and 7 in Tipler