Goals:

1
Lecture 14

• Goals

• Chapter 10
• Understand spring potential energies use
  energy diagrams
• Chapter 11
• Understand the relationship between force,
  displacement and work
• Recognize transformations between kinetic,
  potential, and thermal energies
• Define work and use the work-kinetic energy
  theorem
• Use the concept of power (i.e., energy per time)

• Assignment
• HW7 due Wednesday, Mar. 10
• For Tuesday Read through Ch. 12, Sections 1-3, 5
  6
• Do not concern yourself with the integration
  process in regards to center of mass or moment
  of inertia

2
Energy for a Hookes Law spring

• Associate ½ ku2 with the potential energy of
  the spring

3
Energy for a Hookes Law spring

• Ideal Hookes Law springs are conservative so the
  mechanical energy is constant

4
Energy diagrams

• In general

Ball falling
Spring/Mass system
5
Equilibrium

• Example
• Spring Fx 0 gt dU / dx 0 for xxeq
• The spring is in equilibrium position
• In general dU / dx 0 ? for ANY function
  establishes equilibrium

stable equilibrium
unstable equilibrium
6
Comment on Energy Conservation

• We have seen that the total kinetic energy of a
  system undergoing an inelastic collision is not
  conserved.
• Mechanical energy is lost
• Heat (friction)
• Deformation (bending of metal)
• Mechanical energy is not conserved when
  non-conservative forces are present !
• Momentum along a specific direction is conserved
  when there are no external forces acting in this
  direction.
• Conservation of momentum is a more general
  result than mechanical energy conservation.

7
Mechanical Energy

• Potential Energy (U)
• Kinetic Energy (K)
• If conservative forces
• (e.g, gravity, spring) then
• Emech constant K U
• During ? UspringK1K2 constant Emech
• Mechanical Energy conserved

Before
During
2
1
After
8
Energy (with spring gravity)
Given m, g, h k, how much does the spring
compress?

• Emech constant (only conservative forces)
• At 1 y1 h v1y 0 At 2 y2 0 v2y ?
  At 3 y3 -x v3 0
• Em1 Ug1 Us1 K1 mgh 0 0
• Em2 Ug2 Us2 K2 0 0 ½ mv2
• Em3 Ug3 Us3 K3 -mgx ½ kx2 0
• Given m, g, h k, how much does the spring
  compress?
• Em1 Em3 mgh -mgx ½ kx2 ? Solve ½ kx2
  mgx mgh 0

9
Energy (with spring gravity)
1
mass m
2
h
3
0
-x

• When is the childs speed greatest?
• (A) At y1 (top of jump)
• (B) Between y1 y2
• (C) At y2 (child first contacts spring)
• (D) Between y2 y3
• (E) At y3 (maximum spring compression)

10
Inelastic Processes

• If non-conservative forces (e.g, deformation,
  friction)
• then
• Emech is NOT constant
• After ? K12 lt Emech (before)
• Accounting for this loss we introduce
• Thermal Energy (Eth , new)
• where Esys Emech Eth K U Eth

11
Energy Work

• Impulse (Force vs time) gives us momentum
  transfer
• Work (Force vs distance) tracks energy transfer
• Any process which changes the potential or
  kinetic energy of a system is said to have done
  work W on that system
• DEsys W
• W can be positive or negative depending on the
  direction of energy transfer
• Net work reflects changes in the kinetic energy
• Wnet DK
• This is called the Net Work-Kinetic Energy
  Theorem

12
Circular Motion

• I swing a sling shot over my head. The tension in
  the rope keeps the shot moving at constant speed
  in a circle.
• How much work is done after the ball makes one
  full revolution?

(A) W gt 0
(B) W 0
(C) W lt 0
(D) need more info
13
Examples of Net Work (Wnet)

• DK Wnet
• Pushing a box on a smooth floor with a constant
  force there is an increase in the kinetic energy

Examples of No Net Work

• DK Wnet
• Pushing a box on a rough floor at constant speed
• Driving at constant speed in a horizontal circle
• Holding a book at constant height
• This last statement reflects what we call the
  system
• ( Dropping a book is more complicated because it
  involves changes in U and K, U is transferred to
  K )

14
Changes in K with a constant F

• If F is constant

15
Net Work 1-D Example (constant force)

• A force F 10 N pushes a box across a
  frictionless floor for a distance ?x 5 m.

?x

• Net Work is F ?x 10 x 5 N m 50 J
• 1 Nm 1 Joule and this is a unit of energy
• Work reflects energy transfer

16
Units

• Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
17
Net Work 1-D 2nd Example (constant force)

• A force F 10 N is opposite the motion of a box
  across a frictionless floor for a distance ?x 5
  m.

Finish
Start
q 180
F
?x

• Net Work is F ?x -10 x 5 N m -50 J
• Work reflects energy transfer

18
Work in 3D.

• x, y and z with constant F

19
Work 2-D Example (constant force)

• An angled force, F 10 N, pushes a box across a
  frictionless floor for a distance ?x 5 m and ?y
  0 m

Finish
Start
F
q -45
Fx
?x

• (Net) Work is Fx ?x F cos(-45) ?x 50 x
  0.71 Nm 35 J
• Work reflects energy transfer

20
Scalar Product (or Dot Product)
A B A B cos(q)

• Useful for performing projections.

A ? î Ax î ? î 1 î ? j 0

• Calculation can be made in terms of components.

A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
Calculation also in terms of magnitudes and
relative angles.
A ? B A B cos q
You choose the way that works best for you!
21
Scalar Product (or Dot Product)

• Compare
• A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
• with A as force F, B as displacement Dr
• and apply the Work-Kinetic Energy theorem
• Notice
• F ? Dr (Fx )(Dx) (Fy )(Dz ) (Fz )(Dz)
• Fx Dx Fy Dy Fz Dz DK
• So here
• F ? Dr DK Wnet
• More generally a Force acting over a Distance
  does Work

22
Definition of Work, The basics
Ingredients Force ( F ), displacement ( ? r )
Work, W, of a constant force F acts through a
displacement ? r W F ? r (Work is a scalar)
F
? r
?
displacement
If we know the angle the force makes with the
path, the dot product gives us F cos q and
Dr If the path is curved at each point and
23
Remember that a real trajectory implies forces
acting on an object
path and time
Fradial
Ftang
F


0
Two possible options
0
Change in the magnitude of
Change in the direction of
0

• Only tangential forces yield work!
• The distance over which FTang is applied Work

24
Definition of Work, The basics
Ingredients Force ( F ), displacement ( ? r )
Work, W, of a constant force F acts through a
displacement ? r W F ? r (Work is a scalar)
Work tells you something about what happened on
the path! Did something do work on you? Did
you do work on something? If only one force
acting Did your speed change?
25
ExerciseWork in the presence of friction and
non-contact forces

• A box is pulled up a rough (m gt 0) incline by a
  rope-pulley-weight arrangement as shown below.
• How many forces (including non-contact ones) are
  doing work on the box ?
• Of these which are positive and which are
  negative?
• State the system (here, just the box)
• Use a Free Body Diagram
• Compare force and path

1. 2
2. 3
3. 4
4. 5

26
Lecture 14

• Assignment
• HW7 due Wednesday, March 10
• For Tuesday Read Chapter 12, Sections 1-3, 5 6
• do not concern yourself with the integration
  process