Physics 151: Lecture 16 Today

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Physics 151 Lecture 16Todays Agenda

• Todays Topics
• Conservation of mechanical energy
• Nonconservative forces and loss of energy from a
  system
• Other relationships for potential energy

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Conservation of Energy
See text 8-4

• If only conservative forces are present, the
  total energy (sum of potential and kinetic
  energies) of a system is conserved.

E K U
?E ?K ?U W ?U
using ?K W W (-W) 0
using ?U -W
Both K and U can change, but E K U remains
constant.
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Example The simple pendulum.
See text Example 8.3

• 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.
See text Example 8.3

• 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
y0
v
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Lecture 16, ACT 1The Roller Coaster

• I have built a Roller Coaster . A motor tugs the
  cars to the top and then they are let go and are
  in the hands of gravity. To make the following
  loop, how high do I have to let the release the
  car ??

A) 2R B) 5R C) 5/2 R D) none of the above
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Another Problem Involving a Spring

• A large spring is used to stop the cars after
  they come down the last hill of a roller coaster.
  The cars start at rest at the top of the hill and
  are caught by a mechanism at the instant their
  velocities at the bottom are zero. Compare the
  compression of the spring, xA, for a fully loaded
  car with that, xB, for a lightly loaded car when
  mA 2mB.

a. xA 1/2 xB. b. xA xB. c. xA
(2)1/2 xB. d. xA 2 xB. e. xA 4 xB.
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Non-conservative Forces
See text 8.5

• If the work done does not depend on the path
  taken, the force involved is said to be
  conservative.
• If the work done does depend on the path taken,
  the force involved is said to be
  non-conservative.
• An example of a non-conservative force is
  friction
• 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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Generalized Work Energy Theorem

• WNC ?K ?U ?E

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Lecture 16, ACT 2Stones and Friction

• I throw a stone into the air. While in flight it
  feels the force of gravity and the frictional
  force of the air resistance. The time the stone
  takes to reach the top of its flight path (i.e.
  go up) is,
• A) larger than
• B) equal to
• C) less than
• The time it takes to return from the top (i.e.
  go down).

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Lecture 16, ACT 2Solution

• We know that WNC ?K ?U ?E
• Thus as time progresses the amount of energy in
  the earth-rock system is continually decreasing
• If I consider the way up versus the way down at a
  given height,
• then Eup Kup Uup gt Edown Kdown Udown
• But, at a given height U is always the same.
• So, Kup gt Kdown
• v at a given height is always less on the way
  down

The answer is (C) it takes less time to go up
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Same problem but with numbers

• A 2.0-kg mass is projected vertically upward from
  ground level with an initial speed of 30 m/s. The
  mass rises to a maximum height of 35 m above
  ground level. How much work is done on the mass
  by air resistance between the point of projection
  and the point of maximum height?

a. 0.21 kJ b. 0.21 kJ c. 0.40 kJ d. 0.69
kJ e. 0.69 kJ
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Question

• As an object moves from point A to point B only
  two forces act on it one force is
  nonconservative and does 30 J of work, the other
  force is conservative and does 50 J of work.
  Between A and B,
• the kinetic energy of object increases,
  mechanical energy decreases.
• the kinetic energy of object decreases,
  mechanical energy decreases.
• the kinetic energy of object decreases,
  mechanical energy increases.
• the kinetic energy of object increases,
  mechanical energy increases.
• None of the above.

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Question - 2

• As an object moves from point A to point B only
  two forces act on it one force is conservative
  and does 70 J of work, the other force is
  nonconservative and does 50 J of work. Between A
  and B,
• the kinetic energy of object increases,
  mechanical energy increases.
• the kinetic energy of object decreases,
  mechanical energy increases.
• the kinetic energy of object decreases,
  mechanical energy decreases.
• the kinetic energy of object increases,
  mechanical energy decreases.
• None of the above.

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ACT- 2

• Objects A and B, of mass M and 2M respectively,
  are each pushed a distance d straight up an
  inclined plane by a force F parallel to the
  plane. The coefficient of kinetic friction
  between each mass and the plane has the same
  value . At the highest point,
• K A gt K B .
• K A K B .
• K A lt K B .
• The work done by F on A is greater than the work
  done on B.
• The work done by F on A is less than the work
  done on B.

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Lecture 16, ExampleSkateboard

• Lets now suppose that the surface is not
  frictionless and the same skateboarder reach the
  speed of 7.0 m/s at bottom of the hill. What was
  the work done by friction on the skateboarder ?

Conservation of Total Energy
K1 U1 K2 U2
Wf
m 25 kg
Wf 0 mgR 1/2mv2 0
Wf 1/2mv 2 - mgR
R3 m
Wf (1/2 x25 kg x (7.0 m/s2)2 - - 25 kg
x 10m/s2 3 m)
Wf 613 - 735 J - 122 J
Total mechanical energy decreased by 122 J !
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Conservative Forces and Potential Energy
See Text section 8.6

• We have defined potential energy for conservative
  forces
• DU -W
• But we also now that
• W FxDx
• Combining these two,
• DU - FxDx
• Letting small quantities go to infinitesimals,
• dU - Fxdx
• Or,
• Fx -dU/dx

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Examples of the U - F relationship
See Text section 8.6

• Remember the spring,
• U (1/2)kx2
• Do the derivative
• Fx - d ( (1/2)kx2) / dx
• Fx - 2 (1/2) kx
• Fx -kx

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Examples of the U - F relationship
See Text section 8.6

• Remember gravity,
• Fy mg
• Do the integral
• U -? F dy
• U -? (-mg) dy
• U mgy C
• DU U2 U1 (mgy2 C) (mgy1 C) mgDy

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Recap of todays lecture

• Conservation of Mechanical Energy
• Non-conservative forces and loss of energy from a
  system
• Other relationships for potential energy