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Chapter 6_ Energy

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					      Chapter 6

Conservation of Energy
           Conservation of Energy
            • Work by a Constant Force
            • Kinetic Energy
            • Potential Energy
            • Work by a Variable Force
            • Springs and Hooke’s Law
            • Conservation of Energy
            • Power


MFMcGraw           Ch06 - Energy - Revised: 2/20/10   2
The Law of Conservation of Energy

            The total energy of the Universe is
           unchanged by any physical process.

The three kinds of energy are:
kinetic energy, potential energy, and rest energy.
Energy may be converted from one form to another or
transferred between bodies.



MFMcGraw             Ch06 - Energy - Revised: 2/20/10   3
MFMcGraw   Ch06 - Energy - Revised: 2/20/10   4
       Work by a Constant Force

   Work is an energy transfer by the application of
   a force.
   For work to be done there must be a nonzero
   displacement.

           The unit of work and energy is the joule (J).
                    1 J = 1 Nm = 1 kg m2/s2.



MFMcGraw                 Ch06 - Energy - Revised: 2/20/10   5
                Work - Example
Only the force in the direction of the displacement that does
work.
                                         An FBD for the box at left:
        F
                     Drx                                 y
    q
                                                                         Drx
                                                         N


                                                             q       x
                                                         w       F
                                The work done by the force F is:



MFMcGraw              Ch06 - Energy - Revised: 2/20/10                         6
                Work - Example
The work done by the normal force N is:

The normal force is perpendicular to the displacement.




The work done by gravity (w) is:

The force of gravity is perpendicular to the
displacement.



MFMcGraw              Ch06 - Energy - Revised: 2/20/10   7
                Work - Example

The net work done on the box is:




MFMcGraw            Ch06 - Energy - Revised: 2/20/10   8
                    Work Done
In general, the work done by a force F is defined as




where F is the magnitude of the force, Dr is the magnitude
of the object’s displacement, and q is the angle between F
and Dr (drawn tail-to-tail).




MFMcGraw             Ch06 - Energy - Revised: 2/20/10        9
                    Work - Example
    Example: A ball is tossed straight up. What is the
    work done by the force of gravity on the ball as it
    rises?
                y
                        Dr
 FBD for
 rising ball:
                             x

                    w




MFMcGraw                Ch06 - Energy - Revised: 2/20/10   10
            Inclined Plane-V Constant
A box of mass m is towed up a frictionless incline at constant speed.
The applied force F is parallel to the incline.
                                                                         y
Question: What is the net work done on the box?

 F                                                           F       N
                                                   x
                             An FBD for
                             the box:


                     q                                           q
                                                                     w

            Apply Newton’s 2nd Law:


 MFMcGraw                 Ch06 - Energy - Revised: 2/20/10                   11
           Inclined Plane-V Constant
Example continued:

The magnitude of F is:

If the box travels along the ramp a
distance of Dx the work by the force F is




The work by gravity is




MFMcGraw              Ch06 - Energy - Revised: 2/20/10   12
           Inclined Plane-V Constant
Example continued:

The work by the normal force is:




The net work done on the box is:




MFMcGraw             Ch06 - Energy - Revised: 2/20/10   13
           Inclined Plane-Acceleration
Example: What is the net work done on the box in the
previous example if the box is not pulled at constant speed?




Proceeding as before:
                                                         New Term




MFMcGraw              Ch06 - Energy - Revised: 2/20/10              14
                     Kinetic Energy

                             is an object’s translational
                             kinetic energy.

           This is the energy an object has because of its
                           state of motion.

                  It can be shown that, in general
                      Net Work = Change in K




MFMcGraw                  Ch06 - Energy - Revised: 2/20/10   15
                     Kinetic Energy
Example: The extinction of the dinosaurs and the majority of species on
Earth in the Cretaceous Period (65 Myr ago) is thought to have been
caused by an asteroid striking the Earth near the Yucatan Peninsula. The
resulting ejecta caused widespread global climate change.

  If the mass of the asteroid was 1016 kg (diameter in the range of 4-
  9 miles) and had a speed of 30.0 km/sec, what was the asteroid’s
  kinetic energy?




            This is equivalent to ~109 Megatons of TNT.

 MFMcGraw                  Ch06 - Energy - Revised: 2/20/10              16
           Gravitational Potential Energy
              Part 1- Close to Earth’s Surface

            Potential energy is an energy of position.


    There are potential energies associated with different
   forces. Forces that have a potential energy associated
         with them are called conservative forces.


                  In general


           Not all forces are conservative, i.e. Friction.

MFMcGraw                 Ch06 - Energy - Revised: 2/20/10    17
           Gravitational Potential Energy
The change in gravitational potential energy (only near the
surface of the Earth) is




   where Dy is the change in the object’s vertical position
   with respect to some reference point.
   You are free to choose to location of this where ever it is
   convenient.


MFMcGraw              Ch06 - Energy - Revised: 2/20/10        18
                       GPE Problem
The table is 1.0 m tall and the mass of the box is 1.0 kg.
Ques: What is the change in gravitational potential energy of
the box if it is placed on the table?
                                                             U=0




First: Choose the reference level at the floor. U = 0 here.




MFMcGraw                  Ch06 - Energy - Revised: 2/20/10         19
                     GPE Problem
Example continued:

Now take the reference level (U = 0) to be on top of the
table so that yi = -1.0 m and yf = 0.0 m.




 The results do not
 depend on the location
 of U = 0.



MFMcGraw              Ch06 - Energy - Revised: 2/20/10     20
           Total Mechanical Energy

 Mechanical energy is



 The total mechanical energy of a system is conserved
 whenever nonconservative forces do no work.


           That is   E i = Ef        or DK = -DU.


     Then if DK increases DU decreases and vice versa


MFMcGraw             Ch06 - Energy - Revised: 2/20/10   21
            Mechanical Energy Problem
A cart starts from position 4 with v = 15.0 m/s to the left. Find
the speed of the cart at positions 1, 2, and 3. Ignore friction.




 MFMcGraw               Ch06 - Energy - Revised: 2/20/10       22
           Mechanical Energy Problem
                                                     Or use
                                                     E3=E2




                                                     Or use
                                                     E3=E1
                                                     E2=E1


MFMcGraw          Ch06 - Energy - Revised: 2/20/10            23
              Roller Coaster Problem
A roller coaster car is about to roll down a track. Ignore
friction and air resistance.
                      m = 988 kg


              40
              m                                       20
                                                      m    y=0




(a) At what speed does the
car reach the top of the loop?


 MFMcGraw               Ch06 - Energy - Revised: 2/20/10         24
                Roller Coaster Problem
Example continued:

(b) What is the force exerted on the car by the track at the top
of the loop?
                          Apply Newton’s Second Law:
 FBD for the car:
     y




                 x

    N       w


 MFMcGraw              Ch06 - Energy - Revised: 2/20/10       25
              Roller Coaster Problem
 Example continued:
(c) From what minimum height above the bottom of the track
can the car be released so that it does not lose contact with
the track at the top of the loop?

     Using conservation of mechanical energy:




     Solve for the starting height

 MFMcGraw              Ch06 - Energy - Revised: 2/20/10     26
             Roller Coaster Problem
Example continued:
    What is vmin?    v = vmin when N = 0. This means that




    The initial height must be




MFMcGraw              Ch06 - Energy - Revised: 2/20/10      27
           Nonconservative Forces
What do you do when there are nonconservative forces?
For example, if friction is present




                                        The work done
                                        by friction.




MFMcGraw            Ch06 - Energy - Revised: 2/20/10    28
           Gravitational Potential Energy
            Part 2 - Away from Earth’s Surface


The general expression for gravitational potential energy is:




MFMcGraw              Ch06 - Energy - Revised: 2/20/10      29
           Gravitational Potential Energy

   Example:
   What is the gravitational potential energy of a body of
   mass m on the surface of the Earth?




MFMcGraw              Ch06 - Energy - Revised: 2/20/10       30
                   Planetary Motion
A planet of mass m has an elliptical orbit around the Sun. The
elliptical nature of the orbit means that the distance between the
planet and Sun varies as the planet follows its orbital path. Take
the planet to move counterclockwise from its initial location.
QUES: How does the speed of a planet vary as it orbits the Sun
once?


    The mechanical energy of the
    planet-sun system is:
                                                       B    r    A




 MFMcGraw                Ch06 - Energy - Revised: 2/20/10            31
                    Planetary Motion


                                                      B      r           A


At point “B” the planet is the farthest from the Sun. At point “A” the
planet is at its closest approach to the sun.


 Starting from point “B” (where the planet moves the slowest), as
 the planet moves in its orbit r begins to decrease. As it decreases
 the planet moves faster.
 At point “A” the planet reaches its fastest speed. As the planet
 moves past point A in its orbit, r begins to increase and the planet
 moves slower.
MFMcGraw                  Ch06 - Energy - Revised: 2/20/10                   32
           Work by a Variable Force
           Work can be calculated by finding the area
           underneath a plot of the applied force in the
           direction of the displacement versus the
           displacement.




MFMcGraw                 Ch06 - Energy - Revised: 2/20/10   33
Example: What is the work done by the variable force shown
below?
    Fx (N)
           F3


           F2                                           The net work is then
                                                           W1+W2+W3.
           F1


                x1    x2         x3         x (m)

 The work done by F1 is
 The work done by F2 is
 The work done by F3 is

MFMcGraw             Ch06 - Energy - Revised: 2/20/10                      34
                    Spring Force
By hanging masses on a spring we find that stretch µ
applied force. This is Hooke’s law.

            For an ideal spring: Fx = -kx

Fx is the magnitude of the force exerted by the free end of
the spring, x is the measured stretch of the spring, and k is
the spring constant (a constant of proportionality; its units
are N/m).

           A larger value of k implies a stiffer spring.

MFMcGraw                Ch06 - Energy - Revised: 2/20/10        35
                       Spring Force
(a) A force of 5.0 N applied to the end of a spring cause the
spring to stretch 3.5 cm from its relaxed length.
Ques: How far does a force of 7.0 N cause the same spring
to stretch?

     For springs Fµx. This allows us to write




     Solving for x2:



 MFMcGraw               Ch06 - Energy - Revised: 2/20/10        36
                     Spring Force
Example continued:

  (b) What is the spring constant of this spring?




  Or




MFMcGraw              Ch06 - Energy - Revised: 2/20/10   37
                      Spring Force
An ideal spring has k = 20.0 N/m. What is the amount of work done (by
an external agent) to stretch the spring 0.40 m from its relaxed length?




 MFMcGraw                 Ch06 - Energy - Revised: 2/20/10             38
           Elastic Potential Energy
    The work done in stretching or compressing a spring
    transfers energy to the spring.

    Below is the equation of the spring potential energy.




   The spring is considered the system


MFMcGraw              Ch06 - Energy - Revised: 2/20/10      39
           Elastic Potential Energy
A box of mass 0.25 kg slides along a horizontal, frictionless
surface with a speed of 3.0 m/s. The box encounters a
spring with k = 200 N/m.
Ques: How far is the spring compressed when the box is
brought to rest?




MFMcGraw              Ch06 - Energy - Revised: 2/20/10      40
                           Power

 Power is the rate of energy transfer.


              Average Power

              Instantaneous Power


      The unit of power is the watt. 1 watt = 1 J/s = 1 W.



MFMcGraw               Ch06 - Energy - Revised: 2/20/10      41
           Power - Car Example
A race car with a mass of 500.0 kg completes a quarter-mile
(402 m) race in a time of 4.2 s starting from rest. The car’s
final speed is 125 m/s. (Neglect friction and air resistance.)
Ques: What is the engine’s average power output?




MFMcGraw              Ch06 - Energy - Revised: 2/20/10      42
                      Summary

           • Conservation of Energy
           • Calculation of Work Done by a Constant or
             Variable Force
           • Kinetic Energy
           • Potential Energy (gravitational, elastic)
           • Power




MFMcGraw              Ch06 - Energy - Revised: 2/20/10   43

				
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