Lecture Forces The Laws of Motion by jennyyingdi

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									  Lecture 19:
Introduction to
Solids & Fluids
                 Questions of Yesterday
1) A solid sphere and a hoop of equal radius and mass are both rolled
    up an incline with the same initial velocity. Which object will travel
    farthest up the inclined plane?
    a) the sphere
    b) the hoop
    c) they’ll both travel the same distance up the plane
    d) it depends on the angle of the incline

2) If an acrobat rotates once each second while sailing through the air,
     and then contracts to reduce her moment of inertia to 1/3 of what is
     was, how many rotations per second will result?
     a) once each second
     b) 3 times each second
     c) 1/3 times each second
     d) 9 times each second
                    Practice Problem
 A 10.00-kg cylindrical reel with the radius of 0.500 m and a frictionless
 axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls
into a well, making a light rope unwind from the reel. The bucket starts
                       from rest and falls for 5.00 s.

         10.0 kg               What is the linear acceleration
                                   of the falling bucket?

       0.500 m                      How far does it drop?

                        What is the angular acceleration of the reel?

                     Use energy conservation principles to determine
          5.00 kg                 the speed of the spool
                            after the bucket has fallen 5.00 m
                  4 States of Matter
   Solid           Fluid              Gas              Plasma




   Definite                        Expands to fill    Expands to fill
                     Definite
   Shape &                           any volume        any volume
                     Volume
   Volume                         Takes shape of     Takes shape of
                 Takes shape of
  Molecules                           Container         Container
                    container
close together                    Molecules even     Made up of ions
                   Molecules
    & slow                          farther apart     Fastest of all
                  farther apart
                                   & even faster      matter states
                     & faster
          Density

            Distance between molecules




Density

     M                Density
  r=       The amount of matter (mass)
     V          in a given volume
                       Solids
           Applying force can change shape & size (deform)
           When force is removed -> original shape & size

               Remind you of anything?
Definite                                         x=0

Shape &
               SOLIDS are ELASTIC
Volume
                     FS = -kx

      STRESS = ELASTIC MODULUS x STRAIN

   Force per                              Measure of
                      Elasticity of
   Unit Area                              Deformation
                       material
                Length Elasticity
Elastic Modulus and Induced strain depends on type of stress

                      F                                F
A     L0                               L0

                          F                    DL       F
       L0       DL                      L0

    STRESS = ELASTIC MODULUS x STRAIN


                     F = Y DL
                                               Relative
 SI Units =                                 Length Change
  N/m2 =             A     L0
Pascal (Pa)                             Young’s Modulus
                      LENGTH
              Volume Elasticity

  F                               F               DV


                V0                           V0


      STRESS = ELASTIC MODULUS x STRAIN


                     F = -B DV
                                          Relative
 SI Units =                           Volume Change
  N/m2 =             A       V0
Pascal (Pa)                           Bulk Modulus
                      VOLUME
                          Pressure
Uniform force F is acting over entire
    surface area A in a direction
   perpendicular to surface area
                                        F
          PRESSURE (P)
  Perpendicular Force per unit area
                                                               A
              P= F
                 A
                                            When would this
                                            situation occur?
                          Pressure
Uniform force F is acting over entire
    surface area A in a direction
   perpendicular to surface area
                                         F
          PRESSURE (P)
  Perpendicular Force per unit area
                                                               A
              P= F
                 A
                     Fluids are NOT elastic ->
         do not return to initial state after being deformed
                                 But…
                      Fluids do exerted force
                     Pressure
Force exerted by fluid on a
submerged object is always
   PERPENDICULAR
   to surface of object             F

Fluids exert PRESSURE on
submerged objects and the                                 A
   walls of their container



                Fluids are NOT elastic ->
    do not return to initial state after being deformed
                            But…
                 Fluids do exerted force
                            Pressure
If a fluid is at rest in a container what
                                                      y
          do we know about it?
                                                          0
   It is in EQUILIBRIUM, so…
The net FORCE acting on any                 F1   F2
   portion of fluid is ZERO                               y1


       F1(y1) = -F2(y1)                                   y2

 all points at the same DEPTH
must be at the same PRESSURE

       P1(y1) = P2(y1)
                      Pressure
 The net FORCE acting on any                  y
    portion of fluid is ZERO
                                                  0
 all points at the same DEPTH
                                       P1A
must be at the same PRESSURE
                                                  y1
  ∑Fy = P2A - P1A - Mg = 0
                                                  y2

                           r= M
                              V   Mg    P2A

   P2 = P1 + rg(y1 - y2)
                         Pressure
What is the pressure at the surface
            of the fluid
         (open to the air)?                   P0A        y

   Gas making up atmosphere                                       0
    exerts pressure on fluid
                                                              h

   P2 = P1 + rg(y1 - y2)                                          y1

                                              PA
        P = P0 + rgh

     Pressure P at depth h below the surface of a liquid
 open to the atmosphere is greater than atmosphere pressure
            (P0 = 1.013*105 Pa) by the amount rgh
                         Pressure
What if you change the pressure
   exerted at the surface?
                                       F
        P = P0 + rgh

       PASCAL’S PRINCIPLE
A change in pressure applied to an
    enclosed fluid is transmitted
undiminished to every point of the
fluid and the walls of the container
                        Pressure
                   PASCAL’S PRINCIPLE
A change in pressure applied to an enclosed fluid is transmitted
  undiminished to every point of the fluid and the walls of the
                          container
   What happens if you apply a force F1 to one side of this
                        apparatus?

                 F1
            A1

                                  A2


                                             F2
                        Pressure
                   PASCAL’S PRINCIPLE
A change in pressure applied to an enclosed fluid is transmitted
  undiminished to every point of the fluid and the walls of the
                          container

                         F1/A1 = F2/A2




                F1      Dx1
                                                    Dx2
                                             F2
                        Pressure
                        F1/A1 = F2/A2
                How does Dx1 compare to Dx2?

  Fluids have a definite volume (incompressible) -> DV = 0

                        F1Dx1 = F2Dx2
 What does
this tell you
 about the
 work done       F1     Dx1
                                                  Dx2
on the fluid?                             F2
                       Buoyancy
 What allows an object to float
          in a fluid?

Is the object in equilibrium?
                                  M
                                      V
                       Buoyancy
 What allows an object to float
          in a fluid?                  B

Is the object in equilibrium?
                                   m
   ∑Fy = B - mobjg = 0                     V
                                  mg
                        Buoyancy
  What allows an object to float
           in a fluid?                      P1A

 Is the object in equilibrium?
                                        M
    ∑Fy = B - mobjg = 0                           V
                                   Mg       P2A
∑Fy = P2A - P1A - Mfluidg = 0

B = P2A - P1A = rfluidVfluidg

      BUOYANT
      FORCE
                        Buoyancy
  What allows an object to float
           in a fluid?                             B

 Is the object in equilibrium?
                                               m
    ∑Fy = B - mobjg = 0                                   V
                                          mg
∑Fy = P2A - P1A - Mfluidg = 0
                                        If the object is in
B = P2A - P1A = rfluidVfluidg      equilibrium with the fluid…

      BUOYANT                           robj     Vfluid
                                               =
      FORCE                             rfluid   Vobj
                        Buoyancy
                       B = rfluidVfluidg
             What if the object is rising or sinking?


       B                          B                          B
                                      a                          a
     m                        m                          m
           V                          V                          V
   mg a = 0                mg                           mg

 B = mobjg              B - mobjg > 0           B - mobjg < 0
robj     Vfluid
       =          (rfluid-robj)Vobjg > 0    (rfluid-robj)Vobjg < 0
rfluid   Vobj
      Properties of an Ideal Fluid
                  An Ideal Fluid is…
                       NONVISCOUS
       no internal friction between adjacent layers
                   INCOMPRESSIBLE
                     constant density

               Ideal Fluid Motion is…
                          STEADY
velocity, density, pressure at each point is constant in time

                WITHOUT TURBULENCE
  Angular velocity about center of each element is zero
          All points can translate but not rotate
             Ideal Fluid Motion
How does v1 compare to v2?
                              A2             v2
                                   Dx2

                                Mass is conserved
      A1                v1
                                   DM1 = DM2

             Dx1 = v1Dt          r1A1v1 = r2A2v2
                              Fluid is incompressible
Volume of fluid leaving 1 =
Volume of fluid entering 2         A1v1 = A2v2
 in the same time interval
                              Equation of Continuity
                   Ideal Fluid Motion
                                                 P2A2
 Is energy conserved in an ideal
              fluid?                                v2
                                           Dx2

            P1A1                                     Dy2
                               v1

DM1 = DM2          Dx1   Dy1

               What is the work done on the fluid?
            W1 = P1A1Dx1                 W2 = -P2A2Dx2

                         W = P1V - P2V
          Ideal Fluid Motion
                                           P2A2
W = P1V - P2V
                              A2              v2
                                     Dx2

   P1A1                                       Dy2
                       v1

          Dx1    Dy1

       Is the energy of the fluid changing?
       What types of energy are present?

                Wfluid = DKE + DPE
 P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2
           Ideal Fluid Motion
                                          P2A2
W = P1V - P2V
                               A2            v2
                                    Dx2

    P1A1                                   Dy2
                       v1

           Dx1   Dy1

            BERNOULLI’S EQUATION
The sum of pressure, kinetic energy per unit volume,
        and potential energy per unit volume
     is equal at all points along the streamline
  P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2
                  Questions of the Day
1) Two women of equal mass are standing on the same hard wood
    floor. One is wearing high heels and the other is wearing tennis
    shoes. Which statement is NOT true?
    a) both women exert the same force on the floor
    b) both women exert the same pressure on the floor
    c) the normal force that the floor exerts is the same for both women

2) A boulder is thrown into a deep lake. As the rock sinks deeper and
    deeper into the water what happens to the buoyant force?
    a) it increases
    b) it decreases
    c) it stays the same

								
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