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Kinetic Energy, Work, Power, and Potential Energy

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Title: Kinetic Energy, Work, Power, and Potential Energy


1
Kinetic Energy, Work, Power, and Potential Energy
  • 8.01
  • W05D1

2
Todays Reading Assignment W05D1
  • Young and Freedman 6.1-6.4
  • Math Review Module Scalar Product

3
Kinetic Energy
  • Scalar quantity (reference frame dependent)
  • SI unit is joule
  • Change in kinetic energy

4
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.

5
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

6
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.

7
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.

8
Pushing a Stalled Car

9
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?

10
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

11
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)

12
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?

13
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?
  •  

14
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

15
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

16
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

17
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

18
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.

19
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?

20
Power
  • The average power of an applied force is the rate
    of doing work
  • SI units of power Watts
  • Instantaneous power

21
Work and the Dot Product
22
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

23
Dot Product of Unit Vectors in Cartesian
Coordinates
For unit vectors We have Generally
24
Dot Product in Cartesian Coordinates
25
Kinetic Energy and Dot Product
  • Velocity
  • Kinetic Energy
  • Change in kinetic energy

26
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

27
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
28
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
29
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?

30
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
31
Work Done Along an Arbitrary Path
Work done by force for small displacement Work
done by force along path from A to B
32
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
33
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
34
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.

35
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

36
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.

37
Potential Energy Differnece Constant Gravity
  • Force
  • Work
  • Potential Energy
  • Choice of Zero Point Choose and choose
    . Then
  • Potential Energy

38
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?

39
Worked Example Solution Inverse Square Gravity
  • Force
  • Work done
  • Potential Energy
  • Change
  • Zero Point
  • Potential Energy
  • Function

40
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?

41
Potential Energy Difference Spring Force
  • Force
  • Work done
  • Potential Energy
  • Change
  • Zero Point
  • Potential Energy

42
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

43
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?

44
Next Reading Assignment W05D2
  • Young and Freedman 7.1-7.5,12.3
  • Experiment 3 Energy Transformations
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