Review Exercise 2-3 Solution

 

Section 1: Fill in the Blanks

Note: The correct answer for a blank may require 1 or more than 1 words.

 

n  A basket at rest on a table has 2 forces acting on it, gravity and the normal force. The net force is zero and the basket is said to be in

     _____________________________ equilibrium. 

 

n  The subject concerned with the determination of forces within a structure at rest is called _____________, statics

 

n  Assume that an object is in static equilibrium. Identify the following categories. A slight displacement leads to

 

    Ø  A return to the original position ____________________________  stable

 

    Ø  Further movement away from the original position______________________ unstable

 

    Ø  Rest in the new position__________________________ neutral

 

n  The three common phases of matter are ___________________, ___________________, and _______________________. 

     solid, liquid, gas

 

n  The force due to fluid pressure always acts _______________________ to any solid surface it is in contact with.  perpendicular

 

n  The ratio of the density of a substance to the density of water at 4.00 C is called the __________________ of the substance.  specific

     gravity

 

n  For the derivations in the chapter and the ones I presented an assumption was made concerning the density. It was assumed that the

     fluid is _____________________________ incompressible

 

n  A tire gauge registers __________________ pressure.  To get absolute pressure, you must add _________________________

    pressure to the pressure registered by the tire gauge.  gauge  air

 

n  Atmospheric pressure can be measured with a mercury ______________________ barometer

 

n  The buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object. This is a statement of

    _____________________  principle  Archimedes's

 

n  If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount. This is a

    statement of  _________________________ principle Pascal's

 

n  What 2 assumptions concerning the fluid are used to arrive at Bernoulli's equation?

 

    Ø ______________________________ Incompressible fluid   (also non turbulent flow...)

 

    Ø ______________________________ Viscous forces are small   - will accept any 2 of these

 

n  I presented a derivation in which I began with Bernoulli's equation and its assumption. Two additional assumptions were made to arrive

    at Torricelli's theorem for the exit velocity from a spigot on a cylindrical tank.  List these 2 additional assumptions below.

 

    Ø ______________________________ both top and spigot open to atmospheric pressure

 

    Ø ______________________________ radius of spigot small compared to radius of tank

 

n  The subject of fluids in motion is referred to as fluid dynamics. If the fluid is water, it is referred to as _____________________

     hydrodynamics

 

n   A __________________ is a simple instrument used to measure the specific gravity of a liquid by indicating how deeply the instrument

     sinks in the liquid.   hydrometer

 

n  We can separate fluid flow into 2 major categories. These are

 

    Ø ______________________ flow in which the fluid follows a smooth path.  streamline or laminar

 

    Ø ______________________ flow at higher speeds in which whirlpool-like currents develop.  turbulent

 

n   With the usual definition of terms, the expression r₁A₁v₁ = r₂A₂v₂ is called _______________________ the equation of continuity

 

n   When fluids flow under turbulent conditions, and usually even under non-turbulent conditions, a certain amount of friction develops.

      This friction is referred to as __________________ viscosity

 

n   I derived the expression for velocity of the fluid exiting from a tank through an opening on the side. The expression is

 

     v₂ = ((2gh/(1 - R₂/R₁)⁴)^1/2 .   

 

     In deriving the expression, I started with Bernoulli's equation and did not make one of the assumptions made in deriving Torricelli's

     theorem. The 2 subscript refers to the radius of the spigot or exit and the 1 subscript refers to the radius of the tank.

 

    Ø What is the effect on the exit velocity of increasing the radius of the spigot? ________________ Increases

 

    Ø What principle (sometimes referred to an equation) did I use in deriving the equation? ___________________________ continuity

 

n   F_out/F_in for a hydraulic lift is called the ___________________________ of the hydraulic lift.  mechanical advantage.

 

n   A suction cup will not work in outer space because there is no __________________ to hold the cup in place.  pressure or air

     pressure or difference in air pressure

 

n  Dm/Dt is referred to as a ______________ flow rate.  mass

 

n  The following are all illustrations of ____________________ principle  Bernoulli's

 

    Ø Curve ball

    Ø Chimney smoke

    Ø Atomizer

    Ø Airplane wing

 

 

Section 2 Problems

Give answer (not just numbers in an equation) as a number with appropriate units. If p is involved, leave it  in your answer.

 

n  A 15.0 kg person (person 1, not shown but at the end of the vertical rope below) holds on to a rope attached to a tree. Person 2 (not

    shown) applies a horizontal force and pulls person 1 back a certain distance before releasing. The rope makes an angle of 37⁰ degrees

    with the vertical.

 

        

 

    Ø Draw the free-body diagram for the situation after person 2 applies the force to person 1 but before releasing.  Assume up is the

        positive direction.

 

    Ø What is the magnitude of the force, F, exerted by person 2 on person 1?

 

Apply the first condition of equilibrium to the vertical components of the force and solve for T

 

åF_y = 0        gives T_y - mg = 0

Tsin(53.0 ⁰) - (15.0 kg)(9.8 m/s²) gives

 

T = 184 N

 

Apply the first condition of equilibrium to the horizontal components of the force, solve for F

 

åF_x = 0        gives F = T_x

 

F = Tcos(53.0⁰) gives

 

F = 111 N

 

n  Given the following seesaw arrangement. Assume the following:

 

     w2 is 100 kg

     w1 is 230 kg.

     The total length of the board, d1 + d2, is 30 m

 

     How far from w2 should the fulcrum (the diamond shaped object about which the board pivots) be placed (what is the length of d2)?

 

Approach 1

In the following, the mass of the board is ignored.

SFy = 0   and sum of torque = 0 about any axis. Choose fulcrum as axis to simlify problem as illustrated below.

 

St = 0 gives

 

(mass of w1)g(d1) = (mass of w2)g(d2)

 

d1 = 30 - d2

 

(230 kg)(9.8 m/s²)(30 - d2) = (100 kg)(9.80 m/s²)d2

 

(230 kg)(30 - d2) = (100 kg)(d2)                          dividing both sides by g

 

(230)(30) - 230d2 = 100d2

 

430d2 = (230)(30)    ð      d2 = (230)(30)/330 m

 

Could use sum of forces in y to solve for F_N, normal force at fulcrum but that was not required.

 

Approach 2

In the following, the mass of the board is not ignored. It is assumed to be m and evenly distributed.

The force of gravity on it would be mg.

 

St = 0 with the axis through the board at the  pivot point would give the lever arm for the weight of the board (centered at the pivot) and  F_N would be zero; they would contribute nothing.

 

The torque equation reduces to the one in approach 1 above.

 

n  A solid sphere made of wood has a radius of 0.100 m The mass of the sphere is 1.0 kg.  What is the density of the wood?

 

Density is defined as m/V, so find V      V = (4/3)pr³  = 4.18 x 10^-3 m³           r = 1.0 kg/4.18 x 10^-3 m³  = 239 kg/m³

 

n  An aluminum canoe is floating in a swimming pool. The following is provided:

 

    The density of water is 1.00x10³ kg/m³

    The density of aluminum is 2.70 x 10³ kg/m³,

    The boat displaces 1.00 m³ of water.

 

    Calculate the force of buoyancy, F_Buoyancy, acting on the canoe.

 

(Volume of water displaced)(density of water) = mass of water displaced

 

Mass of water displaced = (1.00 m³)(1.00x10³ kg/m³) = 1000 kg

 

F = mg = (1000 kg)(9.8 m/s²) = 9,800 kg m/s² = 9800 N

 

n  An aluminum canoe is floating in a swimming pool. After a while it begins to leak and sinks to the bottom. Is the water level the same,

     lower, or higher than it was when the canoe was floating?  Indicate your answer by placing a check mark (√) in the appropriate area

     below and clearly explain the rationale for your answer.

 

    Same_____  Lower_____  Higher_____     It is lower

 

The floating canoe displaced a volume of water V whose weight equaled the buoyant force. The sunken canoe, however, displaces a volume of water equal to the volume of its aluminum shell, which is smaller than V.

 

n  A square swimming pool, 10 meters on a side, is 10 meters deep and is filled with water. The density of water is 1.00x10³ kg/m³.

     Ignoring atmospheric pressure, what is the total force on the bottom of the pool?

 

P = F/A  ð F = PA     The area of the bottom of the pool is (10)(10) = 100 m²

 

The pressure at the bottom of the pool is given by P = rgh

 

P = (1.00x10³ kg/m³ )(9.8x10³ m/s²)(10 m) = 98 N/m²

 

F = PA = (98 N/m²)( 100 m²) = 9,800 N

 

 

n  A solid round ball with a diameter of 4 m has a mass of 10 kg uniformly distributed throughout its volume. What is the moment of inertia

    about an axis through its center?

 

I = 2/5Mr² = 2/5(10 kg)(2 m)² = 80/5 = 16 kg m²

 

¢  A 1 kg ball is rotating uniformly at the end of a string in a horizontal circle of 1 meter radius. It makes 1 revolution per second.

     Solve for numerical values.

 

    ð   What is the period, T, of the ball?

 

The ball makes 1 rotation about the center in 1 second, therefore it makes 1 revolution in 1 second. So, T = 1 second.

 

    ð  How fast is the ball travelling?

 

Speed = distance/time= 2 p r/T = 2(3.14)(1)/1 = 6.28 m/s

 

    ð  What is the centripetal acceleration of the ball?

 

a_r = v²/r = (6.28)²/1 = 6.282²

 

Section 3: Multiple Choice

Circle the correct answer - 1 answer per problem

 

n  The first condition for equilibrium is

 

     a. The sum of forces and torques is zero

 

     b. The sum of forces about any axis is zero

 

     c. The sum of forces in each direction ( x, y, and z) is zero   ans

 

     d. The sum of the forces in the x-y plane is zero

 

     e. The sum of the forces in the y-z plane is zero

 

n  The second condition for equilibrium is

 

     a. The sum of the forces acting on an object is zero

 

     b. The sum of the torques acting on an object, as calculated about any axis is zero   ans

 

     c. The sum of the torques acting on an object is zero

 

     d. The sum of the torques in both clockwise and counterclockwise directions must be zero

 

     e. The sum of the torques and moments of inertial must be zero.

 

n  What condition or conditions are necessary for rotational equilibrium?

 

      a.  åF_x = 0

 

      b.  åF_x = 0, åt = 0

 

      c.  åt = 0 about any axis        ans

 

     d.  åF_x = 0, åF_y = 0

 

     e.  åt = 0 about the horizontal axis

 

      

Section 4: Extra Credit

 

n  Rank the following in the order of their birth. A is the one born first, D is the one born last. Partial credit not given:  Rutherford, Galileo,

     Newton, Maxwell.     

 

    A__________________ B__________________ C__________________ D__________________

 

Galileo (1564), Newton (1642), Maxwell (1831), Rutherford (1871)

 

n  Rank the following in the order of their birth. A is the one born first, D is the one born last. Partial credit not given:  Feynman, Dirac,

     Schrödinger, Heisenberg 

 

    A__________________ B__________________ C__________________ D__________________

 

Schrödinger (1887 ), Heisenberg (1901), Dirac (1902), Feynman (1918)

 

n  Two quotations that appear on my site are repeated below. Identify the originator of the quote.

 

    Ø That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe,

         forms my idea of God. ______________   Einstein

 

     Ø The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of

          hypotheses or axioms. ____________  Einstein

 

n  Portraits and short biographies of the major contributors to the material presented in chapter 10 are on my site and are shown below.

     Place the appropriate last name in the space below each portrait.

 

Venturi	Bernoulli	Achimedes	Torricelli	Pascal