8.4 Changes in Mechanical Energy for Nonconservative Forces

1
8.4 Changes in Mechanical Energy for
Nonconservative Forces
2
Mechanical Energy Nonconservative Forces

• If several forces act (conservative
  nonconservative)
• The total work done is Wnet WC WNC
• WC work done by conservative forces
• WNC work done by non-conservative forces
• The work energy principle still holds
• Wnet ?K
• For conservative forces (definition of U)WC
  -?U
• ? ?K -?U WNC
• ? WNC ?K ?U
• Work done by Nonconservative Forces
• Total change in K Total change in U

3
Mechanical Energy Nonconservative Forces, 2

• The work done against friction is greater along
  the red path than along the blue path
• Because the work done depends on the path,
  friction is a nonconservative force

4
Mechanical Energy Nonconservative Forces, 3

• If the work done against the kinetic friction
  depends on the path taken. As the moves moves
  from A to B.
• W kd (8.13)
• d is greater for the curve path so W is greater.
• The work done by the nonconservative force k
  is WNC kd
• Since WNC ?K ?U ?

5
Mechanical Energy Nonconservative Forces, final

• In general, if friction is acting in a system
• DEmech DK DU kd (8.14)
• DU is the change in all forms of potential energy
• If friction is zero, this equation becomes the
  same as Conservation of Mechanical Energy

6
Problem Solving Strategies Nonconservative
Forces

• Define the isolated system and the initial and
  final configuration of the system
• Identify the configuration for zero potential
  energy
• These are the same as for Conservation of Energy
• The difference between the final and initial
  energies is the change in mechanical energy due
  to friction

7
Example 8.6 Motion on a Curve Track
No-conservative Forces (Example 8.7 Text Book)

• A child of mass m starts sliding from rest.
  Frictionless!
• Find speed v at the bottom.
• DEmech DK DU
• DEmech (Kf Ki) (Uf Ui) 0
• (½mvf2 0) (0 mgh) 0
• ½mvf2 mgh 0 ?
• Same result as the child is falling vertically
  trough a distance h!

8
Example 8.6 Motion on a Curve Track, final

• If a kinetic friction acts on the child, find
  DEmech
• Assuming m 20.0 kg and vf 3.00m/s
• DEmech (Kf Ki) (Uf Ui) ? 0
• DEmech ½mvf2 mgh
• DEmech ½(20)(3)2 20(9.8)(2) 302J
• If we want to find ?k
• DEmech 302J
• DEmech kd ?knd 302J ?
• ?k 302/nd

9
Example 8.7 Spring-Mass Collision No-conservative
Forces (Example 8.9 Text Book)

• Frictionless!
• K Us Emech remains constant
• Assuming m 0.80kg vA 1.2m/s k 50N/m
• Find maximum compression of the spring after
  collision (xmax)
• EC EA ? KC UsC KA UsA
• ½mvC2 ½kxmax2 ½mvA2 ½kxA2
• 0 ½kxmax2 ½mvA2 0 ?

10
Example 8.7 Spring-Mass Collision, 2

• If friction is present, the energy decreases by
  DEmech kd
• Assuming ?k 0.50 m 0.80kg vA 1.2m/s k
  50N/m
• Find maximum compression of the spring after
  collision xC
• DEmech k xC ?knxC
• DEmech ?kmgxC ?
• DEmech 3.92xC (1)

11
Example 8.7 Spring-Mass Collision, final

• Using DEmech Ef Ei
• DEmech (Kf Uf) (Ki Ui)
• DEmech 0 ½kxC2 ½mvA2 0
• DEmech 25xC2 0.576 (2)
• Taking (1) (2)
• 25xC2 0.576 3.92xC
• Solving the quadratic equation for xC
• xC 0.092m lt 0.15m (frictionless)
• Expected! Since friction retards the motion of
  the system
• xC 0.25m
• Does not apply since the mass must be to the
  right of the origin.

12
Example 8.8 Connected Blocks in Motion
N-nconservative Forces (Example 8.10 Text Book)

• The system consists of the two blocks, the
  spring, and Earth. Gravitational and potential
  energies are involved
• System is released from rest when spring is
  unstretched.
• Mass m2 falls a distance h before coming to rest.
  
• Find ?k

13
Example 8.8 Connected Blocks in Motion, 2

• The kinetic energy is zero if our initial and
  final configurations are at rest
• Block 2 undergoes a change in gravitational
  potential energy
• The spring undergoes a change in elastic
  potential energy
• ?Emech ?K ?Ug ?US
• ?Emech ?Ug ?US
• ?Emech Ugf Ugf Usf Usi
• ?Emech 0 m2gh ½kh2 0
• ?Emech m2gh ½kh2 (1)

14
Example 8.8 Connected Blocks in Motion, final

• If friction is present, the energy decreases by
  
• DEmech kh ?km1gh (2)
• Taking (1) (2)
• m2gh ½kh2 ?km1gh
• Solving for ?k
• ?km1gh m2gh ½kh2 ?
• ?k (m2gh)/ (m1gh) (½kh2)/(m1gh) ?
• ?k m2/m1 (½kh)/m1g
• This is another way to measure ?k !!!

15
8.5 Conservative Forces and Potential Energy

• NOTE
• We will not cover Section 8.5
• Please Read it!!!
• Here you have a taste

16
Conservative Forces and Potential Energy

• Define a potential energy function, U, such that
  the work done by a conservative force equals the
  decrease in the potential energy of the system
• The work done by such a force, F, is
• (8.15)
• DU is negative when F and x are in the same
  direction

17
Conservative Forces and Potential Energy

• The conservative force is related to the
  potential energy function through
• (8.15)
• The x component of a conservative force acting on
  an object within a system equals the negative of
  the potential energy of the system with respect
  to x

18
Conservative Forces and Potential Energy Check

• Look at the case of a deformed spring
• This is Hookes Law

19
8.6 Energy Diagrams and Equilibrium

• Motion in a system can be observed in terms of a
  graph of its position and energy
• In a spring-mass system example, the block
  oscillates between the turning points, x xmax
• The block will always accelerate back toward x
  0

20
Energy Diagrams and Stable Equilibrium

• The x 0 position is one of stable equilibrium
• Configurations of stable equilibrium correspond
  to those for which U(x) is a minimum
• x xmax and x xmax are called the turning
  points

21
Energy Diagrams and Unstable Equilibrium

• Fx 0 at x 0, so the particle is in
  equilibrium
• For any other value of x, the particle moves away
  from the equilibrium position
• This is an example of unstable equilibrium
• Configurations of unstable equilibrium correspond
  to those for which U(x) is a maximum

22
Neutral Equilibrium

• Neutral equilibrium occurs in a configuration
  when over some region U is constant
• A small displacement from a position in this
  region will produce either restoring or
  disrupting forces

23
Potential Energy in Molecules

• There is potential energy associated with the
  force between two neutral atoms in a molecule
  which can be modeled by the Lennard-Jones
  function

24
Potential Energy Curve of a Molecule

• Find the minimum of the function (take the
  derivative and set it equal to 0) to find the
  separation for stable equilibrium
• The graph of the Lennard-Jones function shows the
  most likely separation between the atoms in the
  molecule (at minimum energy)

25
Force Acting in a Molecule

• The force is repulsive (positive) at small
  separations
• The force is zero at the point of stable
  equilibrium
• The force is attractive (negative) when the
  separation increases
• At great distances, the force approaches zero

26
Material for the Final

• Examples to Read!!!
• Example 8.6 (page 230)
• Example 8.8 (page 232)
• Homework to be solved in Class!!!
• Questions 13, 21
• Problems 36, 45, 48