Goals:

1
Lecture 15

• Goals

• Chapter 11
• Employ conservative and non-conservative forces
• Use the concept of power (i.e., energy per time)
• Chapter 12
• Extend the particle model to rigid-bodies
• Understand the equilibrium of an extended
  object.
• Understand rotation about a fixed axis.
• Employ conservation of angular momentum concept

• Assignment
• HW7 due March 25th
• For Thursday Read Chapter 12, Sections 7-11
• do not concern yourself with the integration
  process in regards to center of mass or moment
  of inertia

2
More 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
• Notice that work reflects energy transfer

3
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

4
Work in 3D.(assigning U to be external to the
system)

• x, y and z with constant F

5
A tool Scalar Product (or Dot Product)
A B A B cos(q)

• Useful for performing projections.

A ? î Ax î ? î 1 î ? j 0
You choose the way that works best for you!
6
Scalar Product (or Dot Product)

• Compare
• A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
• Redefine A ? F (force), B ? Dr (displacement)
• Notice
• F ? Dr (Fx )(Dx) (Fy )(Dz ) (Fz )(Dz)
• So here
• F ? Dr W
• More generally a Force acting over a Distance
  does Work

7
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
8
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

9
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?
10
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

11
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
12
Work and Varying Forces (1D)
Area Fx Dx F is increasing Here W F ? r
becomes dW F dx

• Consider a varying force F(x)

Fx
x
Dx
Finish
Start
F
F
q 0
Dx
Work has units of energy and is a scalar!
13
Example Hookes Law Spring (xi equilibrium)

• 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 (vi) on frictionless surface as shown
  below and xi is the equilibrium position of the
  spring?

14
Example Hookes Law Spring

• How much will the spring compress to bring the
  box to a stop (i.e., v 0 ) if the object is
  moving initially at a constant velocity (vi) on
  frictionless surface as shown below and xe is the
  equilibrium position of the spring? (More
  difficult)

15
Work signs
Notice that the spring force is opposite the
displacement For the mass m, work is
negative For the spring, work is positive
They are opposite, and equal (spring is
conservative)
16
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
17
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

18
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
19
Home 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
20
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
21
Home Exercise Work/Energy for Non-Conservative
Forces

• The air track is once again at an angle of 30
  with respect to horizontal. The cart (with mass 1
  kg) is released 1 meter from the bottom and hits
  the bumper at a speed, v1. This time the vacuum/
  air generator breaks half-way through and the air
  stops. The cart only bounces up half as high as
  where it started.
• How much work did friction do on the cart ?(g10
  m/s2)

1 meter
30
(A) 2.5 J (B) 5 J (C) 10 J (D) 2.5 J (E)
5 J (F) 10 J
22
Home Exercise Work/Energy for Non-Conservative
Forces

• How much work did friction do on the cart ? (g10
  m/s2)
• W F Dx is not easy to do
• Work done is equal to the change in the energy of
  the system (U and/or K). Efinal - Einitial and
  is lt 0. (E UK)
• Use W Ufinal - Uinit mg ( hf - hi ) - mg
  sin 30 0.5 m
• W -2.5 N m -2.5 J or (D)

hi
hf
1 meter
30
(A) 2.5 J (B) 5 J (C) 10 J (D) 2.5 J (E)
5 J (F) 10 J
23
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 !

24
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) .

25
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
26
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.

27
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)

28
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

29
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,

1. Top
2. Middle
3. Bottom

30
Chap. 12 Rotational Dynamics

• Up until now rotation has been only in terms of
  circular motion with ac v2 / R and aT d
  v / dt
• Rotation is common in the world around us.
• Many ideas developed for translational motion are
  transferable.

31
Conservation of angular momentum has consequences
How does one describe rotation (magnitude and
direction)?
32
Rotational Dynamics A childs toy, a physics
playground or a students nightmare

• A merry-go-round is spinning and we run and jump
  on it. What does it do?
• We are standing on the rim and our friends spin
  it faster. What happens to us?
• We are standing on the rim a walk towards the
  center. Does anything change?

33
Rotational Variables

• Rotation about a fixed axis
• Consider a disk rotating aboutan axis through
  its center
• How do we describe the motion
• (Analogous to the linear case )

34
Rotational Variables...

• Recall At a point a distance R away from the
  axis of rotation, the tangential motion
• x ? R
• v ? R
• a ? R

35
Overview (with comparison to 1-D kinematics)

• Angular Linear

And for a point at a distance R from the rotation
axis
x R ????????????v ? R ??????????aT ? R
Here aT refers to tangential acceleration
36
Exercise Rotational Definitions

• A friend at a party (perhaps a little tipsy) sees
  a disk spinning and says Ooh, look! Theres a
  wheel with a negative w and positive a!
• Which of the following is a true statement about
  the wheel?

1. The wheel is spinning counter-clockwise and
   slowing down.
2. The wheel is spinning counter-clockwise and
   speeding up.
3. The wheel is spinning clockwise and slowing down.
4. The wheel is spinning clockwise and speeding up

37
Lecture 15

• Assignment
• HW7 due March 25th
• For Thursday Read Chapter 12, Sections 7-11
• do not concern yourself with the integration
  process in regards to center of mass or moment
  of inertia