Submerged Surface and Equilibrium

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Submerged Surface and Equilibrium

• ENGR 221
• February 12, 2003

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Lecture Goals

• 5.7 Forces on Submerged Surfaces
• 6-2 Free-Body Diagrams
• 6-3 Equilibrium in Two Dimensions
• 6-4 Equilibrium in Three Dimensions

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Quiz
Determine the moment about the line AB.
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Force on Submerged Surfaces
In a fluid at rest, the weight of the liquid will
create a pressure on the surface of a body. This
pressure is defined as the hydrostatic pressure.
where PA is pressure absolute, P0 is the
initial pressure and g is the specific weight of
the fluid in F/L3 and d is the depth.
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Force on Submerged Surfaces
The density of fluid, r is multiplied by the g to
get the specific weight of the fluid and PG
(gauge pressure) is defined as.
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Force on a Submerged Surface
The pressure acts as a function of depth.
R
The resultant force, R
gd
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Force on a Submerged Surface
How does on find the forces on a submerge surface
at an angle?
Draw the the free-body diagram.
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Force on a Submerged Surface
The free-body diagram would have
The pressure and the weight of the fluid.
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Force on a Submerged Surface
The resulting force distribution without the
weight of the water would look like,
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Force on a Submerged Surface
If we were take a look at the distribution on a
non-linear surface the results would
The force can be represented as
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Force on Submerged Surface - Example
The quick action gate AB is 1.75 ft wide and is
held in it closed position by a vertical cable
and by hinges located along its top edge B. For
a depth of water d 6-ft determine the force
acting on the gate and location of the force.
Determine the tension in the rope to maintain
keep the gate closed.
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Force on Submerged Surface - Example
The pressure acting on the wall is equivalent to
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Force on Submerged Surface - Example
The pressure acting on the wall is equivalent to
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Force on Submerged Surface - Example
The force in the x and y direction are
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Force on Submerged Surface - Example
The location of the force is
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Force on Submerged Surface - Example
If we look at were that is located it is
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Force on Submerged Surface - Example

The resultant force is applied at
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Force on Submerged Surface - Example

An alternative is to apply the same loading over
the gate which has length of
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Force on Submerged Surface - Example

An alternative is to apply the same loading over
the gate which is
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Equilibrium
As with equilibrium of forces for static
systems. In order for the body to be at rest the
moment of the body must be in equilibrium so that
the total moment of a system is zero.
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Equilibrium
Therefore, if the total moment is defined
as Therefore the resulting scalar components of
the system will have to be
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Free Body Diagrams
The first step in solving a problem is drawing a
free-body diagram (FBD). This step is the most
crucial and important solving any problem. It
defines weight of the body, the known external
forces, and unknown external forces. It defines
the constraints and the directions of the forces.
If the FBD is drawn correctly the solving of the
problem is trivial.
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Free Body Diagrams
Construction of a free body diagram.
Step 1 Step 2 Step 3
Decide which body or combination of bodies are to
be shown on the free-body diagram. Prepare
drawing or sketch of the outline of the isolated
or free body. Carefully trace around the boundary
of the free-body and identify all the forces
exerted by contacting or attracting bodies that
were removed during isolation process.
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Free Body Diagrams
Construction of a free body diagram(cont.)
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Connections
Reaction Equivalent to a Force with Known Line of
Action Number of Unknowns 1
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Connections
Reaction Equivalent to a Force with Known Line of
Action Number of Unknowns 1
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Connections
Reaction Equivalent to a Force with Known Line of
Action Number of Unknowns 1
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Connections
Reactions Equivalent to a Force with Unknown
Direction Number of Unknowns 2
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Connections
Reactions Equivalent to a Force with Unknown
Direction and a couple Number of Unknowns 3
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Connections
Additional connection is
Linear elastic spring with a spring constant, k.
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Connections
Additional connection is
For static loads, the tension in the line of the
ideal pulley is equal. T1T2
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Connections
Additional connection is
Set of slides with pin and fixed connection.
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Free-Body Diagram - Example
Draw the free-body diagram of the bar AB.
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Free-Body Diagram - Example
The results
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Free-Body Diagram - Example
Draw the free-body diagram of the cart.
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Free-Body Diagram - Example
Results are
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Free-Body Diagram - Class
Draw the free-body diagram for the cylinder.
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Free-Body Diagram - Class
Draw the free-body diagram for the rod DF and
BCE.
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Equilibrium Example (I)
Determine the reaction at the on the beam due to
the loading.
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Equilibrium Example (I)
Draw the free-body diagram
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Equilibrium Example (I)
Look at the sum of the forces, 3 unknowns.
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Equilibrium Example (I)
The third equation is the summation of
moments. Selection taking the moment at A
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Equilibrium Example (I)
Back substitute into the equations.
The results are
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Equilibrium Example (II)
A rope and pulley system is used to support a
body. Each pulley is free to rotate, and the
ropes are continuous over the pulley Determine
the force P required to hold the body in
equilibrium if the mass m of the body is 250 kg.
Assume pulleys are mass-less.
C
B
A
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Equilibrium Example (II)
Draw the free-bodies of the pulleys.
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Equilibrium Example (II)
Use Pulley B
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Equilibrium Example (II)
Use Pulley C, the load P1 is translated to pulley
A
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Equilibrium Class Problem
Determine the reactions at A and B due to the
loading.
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Equilibrium Example (III)
Determine the reactions at B and the tension in
cable at A due to the loading.
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Equilibrium Example (III)
Draw the free-body diagram of bar ABC. Note that
the cable is in tension, however the reactions at
B may or may not be correct directions.
B
A
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Equilibrium Example (III)
Use the summation of the forces.
B
A
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Equilibrium Example (III)
Use the lines of action and moment about B
B
A
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Equilibrium Example (III)
B
Back substitution
A
The results are
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Equilibrium Class Problem
Determine the reactions at A and the force in bar
CD due to the loading.
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Equilibrium in 3-Dimensions
In two dimensions, the equations are solved using
the summation of forces in the x, y and z
directions and the moment equilibrium includes
moment components in the x, y and z directions.
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Homework (Due 2/19/03)
Problems
5-35, 5-37, 5-40, 5-44, 5-69, 5-72, 5-85
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Equilibrium Bonus Problem
Determine the reactions at A and the reaction at
E due to the loading. Not the weight of the
structure.