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
• HW6 due Wednesday, Mar. 11
• For Tuesday Read Chapter 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 kx2 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
Emech
K
Energy
U
x
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)
• Bending of metal and deformation
• Kinetic energy is not conserved by these
  non-conservative forces occurring during the
  collision !
• Momentum along a specific direction is conserved
  when there are no external forces acting in this
  direction.
• In general, easier to satisfy conservation of
  momentum than 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)
1
2
h
3
0
mass m
-x

• 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
Energy (with spring gravity)
1
2
h
3
kx
mg
0
-x

• When is the childs speed greatest?
• A Calculus soln. Find v vs. spring displacement
  then maximize
• (i.e., take derivative and then set to zero)
• B Physics As long as Fgravity gt Fspring then
  speed is increasing
• Find where Fgravity- Fspring 0 ? -mg
  kxVmax or xVmax -mg / k
• So mgh Ug23 Us23 K23 mg (-mg/k) ½
  k(-mg/k)2 ½ mv2
• ? 2gh 2(-mg2/k) mg2/k v2 ? 2gh mg2/k
  vmax2

11
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

12
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

13
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
14
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 )

15
Changes in K with a constant F

• If F is constant

16
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

17
Units

• Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
18
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

19
Work in 3D.

• x, y and z with constant F

20
Work 2-D Example (constant force)

• A 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

21
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!
22
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

23
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
24
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

25
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?
26
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?
• Use a Free Body Diagram
• Compare force and path

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

27
Home ExerciseWork Done by Gravity

• An frictionless track is at an angle of 30 with
  respect to the horizontal. A cart (mass 1 kg) is
  released from rest. It slides 1 meter downwards
  along the track bounces and then slides upwards
  to its original position.
• How much total work is done by gravity on the
  cart when it reaches its original position? (g
  10 m/s2)

1 meter
30
(A) 5 J (B) 10 J (C) 20 J (D) 0 J
28
Lecture 14

• Assignment
• HW6 due Wednesday, March 11
• For Tuesday Read Chapter 12, Sections 1-3, 5 6
• do not concern yourself with the integration
  process