Kinetic Energy, Work, Power, and Potential Energy

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Kinetic Energy, Work, Power, and Potential Energy

• 8.01
• W05D1

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Todays Reading Assignment W05D1

• Young and Freedman 6.1-6.4
• Math Review Module Scalar Product

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Kinetic Energy

• Scalar quantity (reference frame dependent)
• SI unit is joule
• Change in kinetic energy

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Concept Question Work and Kinetic Energy

• Compared to the amount of energy required to
  accelerate a car from rest to 10 mph (miles per
  hour), the amount of energy required to
  accelerate the same car from 10 mph to 20 mph is
• (1) the same
• (2) twice as much
• (3) three times as much
• (4) four times as much
• (5) unsure.

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Work Done by a Constant Force for One Dimensional
Motion

• Definition
• The work W done by a constant force with an
  x-component, Fx, in displacing an object by ?x
  is equal to the x-component of the force times
  the displacement

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Concept Question Pushing against a wall

• The work done by the contact force of the wall
  on the person as the person moves away from the
  wall is
• positive.
• negative.
• zero.
• impossible to determine from information given in
  question and the figure.

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Concept Question Work and Walking

• When a person walks, the force of friction
  between the floor and the person's feet
  accelerates the person forward. The work done by
  the friction force is
• positive.
• negative.
• zero.

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Pushing a Stalled Car

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Table Problem Work Done by Gravity Near the
Surface of the Earth

• Consider an object of mass m near the surface
  of the earth falling directly towards the center
  of the earth. The gravitational force between the
  object and the earth is nearly constant. Suppose
  the object starts from an initial point that is a
  distance y0 from the surface of the earth and
  moves to a final point a distance yf from the
  surface of the earth. How much work does the
  gravitational force do on the object as it falls?

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Work done by Non-Constant Force One Dimensional
Motion

• (Infinitesimal) work is a scalar
• Add up these scalar quantities to get the total
  work as area under graph of Fx vs x

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Concept Question Work due to Variable Force

• A particle starts from rest at x 0 and moves
  to x L under the action of a variable force
  F(x), which is shown in the figure. What is the
  particle's kinetic energy at x L/2 and at x
  L?
• (Fmax)(L/2), (Fmax)(L)
• (2) (Fmax)(L/4), 0
• (3) (Fmax)(L), 0
• (4) (Fmax)(L/4), (Fmax)(L/2)
• (5) (Fmax)(L/2), (Fmax)(L/4)

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Table Problem Work Done by the Spring Force

• Connect one end of a spring of length l0 with
  spring constant k to an object resting on a
  smooth table and fix the other end of the spring
  to a wall. Stretch the spring until it has length
  l and release the object.
• How much work does the spring do on the object
  as a function of x l - l0, the distance the
  spring has been stretched or compressed?

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Worked Example Work Done by Several Forces

• A block of mass m slides along a horizontal
  table with speed v0. At x 0 it hits a spring
  with spring constant k and begins to experience a
  friction force. The coefficient of kinetic
  friction is given by m. How far did the spring
  compress when the block first momentarily comes
  to rest?
•  

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Recall integration formula for acceleration with
respect to time

• The x-component of the acceleration of an object
• is the derivative of the x-component of the
  velocity
• Therefore the integral of x-component of the
  acceleration with respect to time, is the
  x-component of the velocity

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Integration formula for acceleration with respect
to displacement

• The integral of x-component of the acceleration
  with respect to the displacement of an object, is
  given by
• Multiply both sides by the mass of the object
  giving integration formula

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Work-Kinetic Energy Theorem One Dimensional Motion

• Substitute Newtons Second Law (in one dimension)
  
• in definition of work integral which then becomes
• Apply integration formula to get work-kinetic
  energy theorem

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Concept Question Work-Energy
An object is dropped to the earth from a height
of 10m. Which of the following sketches best
represent the kinetic energy of the object as it
approaches the earth (neglect friction)?

1. a
2. b
3. c
4. d
5. e

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Concept Question

• Two objects are pushed on a frictionless
  surface from a starting line to a finish line
  with equal constant forces. One object is four
  times as massive as the other. Both objects are
  initially at rest. Which of the following
  statements is true when the objects reach the
  finish line?
• The kinetic energies of the two objects are
  equal.
• Object of mass 4m has the greater kinetic energy.
  
• Object of mass m has the greater kinetic energy.
• Not information is given to decide.

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Worked Example Work-Energy Theorem for Inverse
Square Gravitational Force

• Consider a magnetic rail gun that shoots an
  object of mass m radially away from the surface
  of the earth (mass me). When the object leaves
  the rail gun it is at a distance ri from the
  center of the earth moving with speed vi . What
  speed of the object as a function of distance
  from the center of the earth?

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Power

• The average power of an applied force is the rate
  of doing work
• SI units of power Watts
• Instantaneous power

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Work and the Dot Product
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Dot Product

• A scalar quantity
• Magnitude
• The dot product can be positive, zero, or
  negative
• Two types of projections the dot product is the
  parallel component of one vector with respect to
  the second vector times the magnitude of the
  second vector

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Dot Product of Unit Vectors in Cartesian
Coordinates
For unit vectors We have Generally
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Dot Product in Cartesian Coordinates
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Kinetic Energy and Dot Product

• Velocity
• Kinetic Energy
• Change in kinetic energy

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Work Done by a Constant Force

• Definition Work
• The work done by a constant force on
  an object is equal to the component of the force
  in the direction of the displacement times the
  magnitude of the displacement
• Note that the component of the force in the
  direction of the displacement can be positive,
  zero, or negative so the work may be positive,
  zero, or negative

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Concept Question Work and Gravity 1

• A ball is given an initial horizontal velocity
  and allowed to fall under the influence of
  gravity near the surface of the earth, as shown
  below. The work done by the force of gravity on
  the ball is

(1) positive (2) zero (3) negative
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Worked Example Work Done by a Constant Force in
Two Dimensions
Force exerted on the object Components Cons
ider an object undergoing displacement Work
done by force on object
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Table Problem Work Constant Forces and Dot
Product
An object of mass m, starting from rest, slides
down an inclined plane of length s. The plane is
inclined by an angle of ? to the ground. The
coefficient of kinetic friction is µ.

• Use the dot product definition of work to
  calculate the work done by the normal force, the
  gravitational force, and the friction force as
  the object displaces a distance s down the
  inclined plane.
• For each force, is the work done by the force
  positive or negative?
• What is the sum of the work done by the three
  forces? Is this positive or negative?

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Concept Question Work and inverse square gravity

• A comet is speeding along a hyperbolic orbit
  toward the Sun. While the comet is moving away
  from the Sun, the work done by the Sun on the
  comet is

(1) positive (2) zero (3) negative
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Work Done Along an Arbitrary Path
Work done by force for small displacement Work
done by force along path from A to B
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Work-Energy Theorem in Three-Dimensions
As you will show in the problem set, the one
dimensional work-kinetic energy theorem
generalizes to three dimensions
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Work Path Dependent Line Integral
Work done by force along path from A to B
In order to calculate the line integral, in
principle, requires a knowledge of the path.
However we will consider an important class of
forces in which the work line integral is
independent of the path and only depends on the
starting and end points
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Conservative Forces

• Definition Conservative Force If the work done
  by a force in moving an object from point A to
  point B is independent of the path (1 or 2),
• then the force is called a conservative force
  which we denote by . Then the work done only
  depends on the location of the points A and B.

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Example Gravitational Force

• Consider the motion of an object under the
  influence of a gravitational force near the
  surface of the earth
• The work done by gravity depends only on the
  change in the vertical position

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Potential Energy Difference

• Definition Potential Energy Difference between
  the points A and B associated with a conservative
  force is the negative of the work done by
  the conservative force in moving the body along
  any path connecting the points A and B.

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Potential Energy Differnece Constant Gravity

• Force
• Work
• Potential Energy
• Choice of Zero Point Choose and choose
  . Then
• Potential Energy

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Worked Example Change in Potential Energy for
Inverse Square Gravitational Force

• Consider an object of mass m1 moving towards
  the sun (mass m2). Initially the object is at a
  distance r0 from the center of the sun. The
  object moves to a final distance rf from the
  center of the sun. For the object-sun system,
  what is the change in potential during this
  motion?

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Worked Example Solution Inverse Square Gravity

• Force
• Work done
• Potential Energy
• Change
• Zero Point
• Potential Energy
• Function

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Table Problem Change in Potential Energy Spring
Force

• Connect one end of a spring of length l0 with
  spring constant k to an object resting on a
  smooth table and fix the other end of the spring
  to a wall. Stretch the spring until it has length
  l and release the object. Consider the
  object-spring as the system. When the spring
  returns to its equilibrium length what is the
  change in potential energy of the system?

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Potential Energy Difference Spring Force

• Force
• Work done
• Potential Energy
• Change
• Zero Point
• Potential Energy

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Work-Energy Theorem Conservative Forces

• The work done by the force in moving an object
  from A to B is equal to the change in kinetic
  energy
• When the only forces acting on the object are
  conservative forces
• then the change in potential energy is
• Therefore

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Table Problem Asteroid about Sun

• An asteroid of mass m is in a non-circular
  closed orbit about the sun. Initially it is a
  distance ri from the sun, with speed vi. What is
  the change in the kinetic energy of the asteroid
  when it is a distance is rf, from the sun?

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Next Reading Assignment W05D2

• Young and Freedman 7.1-7.5,12.3
• Experiment 3 Energy Transformations