Physics 207, Lecture 15, Oct. 22

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Physics 207, Lecture 15, Oct. 22

• 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 Oct. 29
• For Monday Read Chapter 12, Sections 7, 8 11
• do not concern yourself with the integration
  process in regards to center of mass or moment
  of inertia

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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!
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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?

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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)

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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)
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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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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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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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Home Exercise Work Friction

• Two blocks having mass m1 and m2 where m1 m2.
  They are sliding on a frictionless floor and have
  the same kinetic energy when they encounter a
  long rough stretch (i.e. m 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 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
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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
• 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
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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 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 path1 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 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

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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.

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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?

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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 )

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Rotational Variables...

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

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Overview (with comparison to 1-D kinematics)

• Angular Linear

distance R from the rotation And for a point at a
axis
x R ????????????v ? R ??????????a ? R
Here the a refers to tangential acceleration
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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?

• The wheel is spinning counter-clockwise and
  slowing down.
• The wheel is spinning counter-clockwise and
  speeding up.
• The wheel is spinning clockwise and slowing down.
• The wheel is spinning clockwise and speeding up

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Example Wheel And Rope

• A wheel with radius r 0.4 m rotates freely
  about a fixed axle. There is a rope wound around
  the wheel.
• Starting from rest at t 0, the rope is
  pulled such that it has a constant acceleration
  aT 4m/s2.
• How many revolutions has the wheel made after
  10 seconds?
• (One revolution 2? radians)

aT
r
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Example Wheel And Rope

• A wheel with radius r 0.4 m rotates freely
  about a fixed axle. There is a rope wound around
  the wheel. Starting from rest at t 0, the rope
  is pulled such that it has a constant
  acceleration aT 4 m/s2. How many revolutions
  has the wheel made after 10 seconds?
  (One revolution 2? radians)
• Revolutions R (q - q0) / 2p and aT a r
• q q0 w0 Dt ½ a Dt2 ?
• R (q - q0) / 2p 0 ½ (a/r) Dt2 / 2p
• R (0.5 x 10 x 100) / 6.28

aT
r
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System of Particles (Distributed Mass)

• Until now, we have considered the behavior of
  very simple systems (one or two masses).
• But real objects have distributed mass !
• For example, consider a simple rotating disk and
  2 equal mass m plugs at distances r and 2r.
• Compare the velocities and kinetic energies at
  these two points.

w
1
2
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System of Particles (Distributed Mass)

• The rotation axis matters too!
• Twice the radius, four times the kinetic energy

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System of Particles Center of Mass (CM)

• If an object is not held then it will rotate
  about the center of mass.
• Center of mass Where the system is balanced !
• Building a mobile is an exercise in finding
• centers of mass.

mobile
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System of Particles Center of Mass

• How do we describe the position of a system
  made up of many parts ?
• Define the Center of Mass (average position)
• For a collection of N individual pointlike
  particles whose masses and positions we know

RCM
m2
m1
r2
r1
y
x
(In this case, N 2)
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Sample calculation

• Consider the following mass distribution

XCM (m x 0 2m x 12 m x 24 )/4m meters YCM
(m x 0 2m x 12 m x 0 )/4m meters XCM 12
meters YCM 6 meters
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System of Particles Center of Mass

• For a continuous solid, convert sums to an
  integral.

dm
r
y
where dm is an infinitesimal mass element.
x
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Connection with motion...

• So for a solid object which rotates about its
  center of mass and whose CM is moving

VCM
?
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Rotational Dynamics What makes it spin?
A force applied at a distance from the rotation
axis gives a torque
a
FTangential
F
Frandial

• ?TOT r FTang r F sin f

r

• Only the tangential component of the force
  matters. With torque the position of the force
  matters
• Torque is the rotational equivalent of force
• Torque has units of kg m2/s2 (kg m/s2) m N m
• A constant torque gives constant angular
  acceleration iff the mass distribution and the
  axis of rotation remain constant.

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Physics 207, Lecture 15, Oct. 22

• Assignment
• HW7 due Oct. 29
• For Wednesday Read Chapter 12, Sections 7, 8
  11
• do not concern yourself with the integration
  process in regards to center of mass or moment
  of inertia