Physics 207, Lecture 14, Oct' 20

1
Physics 207, Lecture 14, Oct. 20

• Goals

• 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
• Develop a complete statement of the law of
  conservation of energy
• Use the concept of power (i.e., energy per time)

• Assignment
• HW6 due Wednesday, Oct. 22, HW7 available today
• For Wednesday 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

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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
3
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

4
Energy (with spring gravity)
1
mass m
2
h
3
0
-x

• When is the childs speed greatest?
• (A) At y1
• (B) Between y1 y2
• (C) At y2
• (D) Between y2 y3
• (E) At y3

5
Energy (with spring gravity)
1
2
h
3
kx
mg
0
-x

• When is the childs speed greatest? (D) Between
  y2 y3
• 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

6
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

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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

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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
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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 )

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Changes in K with a constant F

• In 1-dimension,
• F ma m dv/dt m dv/dx dx/dt m dv/dx v
• by the chain rule so that F dx mv dv

• If F is constant

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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

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Units

• Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
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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

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Work in 3D.

• x, y and z with constant F

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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

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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!
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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

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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
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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 are important one for
  work!
• Direct application of Newtons Laws gives the
  kinematics
• The distance over which FTang is applied Work

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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?
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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

• 2
• 3
• 4
• 5

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Exercise Work 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 are doing work on the box ?
• And which are positive (T) and which are
  negative (f, mg)?
• (For mg only the component along the surface is
  relevant)
• Use a Free Body Diagram

• (A) 2
• (B) 3 is correct
• 4
• 5

v
N
T
f
mg
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Physics 207, Lecture 14, Oct. 20

• Assignment
• HW6 due Wednesday, Oct. 22
• HW7 available today
• For Wednesday Read Chapter 12, Sections 1-3, 5
  6
• do not concern yourself with the integration
  process
• Next slides are on Wednesday

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Work and Varying Forces (1D)

• Consider a varying force F(x)

Area Fx Dx F is increasing Here W F ? r
becomes dW F dx
Fx
x
Dx
Finish
Start
F
F
q 0
Dx
Work has units of energy and is a scalar!
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Work Kinetic-Energy Theorem

• Net Work done change in kinetic energy

(final initial)
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Example Work Kinetic-Energy Theorem with
variable force

• How much will the spring compress (i.e. ?x) to
  bring the box to a stop (i.e., v 0 ) if the
  object is moving initially at a constant velocity
  (vo) on frictionless surface as shown below ?

Notice that the spring force is opposite the
displacement For the mass m, work is
negative For the spring, work is positive
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Example Work Kinetic-Energy Theorem with
variable force

• How much will the spring compress (i.e. ?x xf -
  xi) to bring the box to a stop (i.e., v 0 ) if
  the object is moving initially at a constant
  velocity (vo) on frictionless surface as shown
  below ?

vo
to
m
spring at an equilibrium position
?x
F
V0
t
m
spring compressed
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Conservative Forces Potential Energy

• For any conservative force F we can define a
  potential energy function U in the following way
• The work done by a conservative force is equal
  and opposite to the change in the potential
  energy function.
• This can be written as

ò
W F dr - ?U
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Exercise Work Friction

• Two blocks having mass m1 and m2 where m1 gt m2.
  They are sliding on a frictionless floor and have
  the same kinetic energy when they encounter a
  long rough stretch (i.e. m gt 0) which slows them
  down to a stop.
• Which one will go farther before stopping?
• Hint How much work does friction do on each
  block ?

(A) m1 (B) m2 (C) They will go the same
distance
m1
v1
v2
m2
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Exercise Work Friction

• W F d - m N d - m mg d DK 0 ½ mv2
• - m m1g d1 - m m2g d2 ? d1 / d2 m2 / m1

(A) m1 (B) m2 (C) They will go the same
distance
m1
v1
v2
m2
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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
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Conservative Forces and Potential Energy

• So we can also describe work and changes in
  potential energy (for conservative forces)
• DU - W
• Recalling (if 1D)
• W Fx Dx
• Combining these two,
• DU - Fx Dx
• Letting small quantities go to infinitesimals,
• dU - Fx dx
• Or,
• Fx -dU / dx

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Non-conservative Forces

• 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.
• and work done is proportional to the length of
  the path !

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A Non-Conservative Force, Friction

• Looking down on an air-hockey table with no air
  flowing (m gt 0).
• Now compare two paths in which the puck starts
  out with the same speed (Ki path 1 Ki path 2) .

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A Non-Conservative Force
Since path2 distance gtpath1 distance the puck
will be traveling slower at the end of path 2.
Work done by a non-conservative force
irreversibly removes energy out of the system.
Here WNC Efinal - Einitial lt 0 ? and
reflects Ethermal
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Work Power

• Two cars go up a hill, a Corvette and a ordinary
  Chevy Malibu. Both cars have the same mass.
• Assuming identical friction, both engines do the
  same amount of work to get up the hill.
• Are the cars essentially the same ?
• NO. The Corvette can get up the hill quicker
• It has a more powerful engine.

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Work Power

• Power is the rate at which work is done.
• Average Power is,
• Instantaneous Power is,
• If force constant, W F Dx F (v0 Dt ½ aDt2)
• and P W / Dt F (v0 aDt)

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Work Power

• Power is the rate at which work is done.

Units (SI) are Watts (W)
Instantaneous Power
Average Power
1 W 1 J / 1s
Example

• A person of mass 80.0 kg walks up to 3rd floor
  (12.0m). If he/she climbs in 20.0 sec what is
  the average power used.
• Pavg F h / t mgh / t 80.0 x 9.80 x 12.0 /
  20.0 W
• P 470. W

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Exercise Work Power

• Starting from rest, a car drives up a hill at
  constant acceleration and then suddenly stops at
  the top.
• The instantaneous power delivered by the engine
  during this drive looks like which of the
  following,

• Top
• Middle
• Bottom