Overview (Welcome Parents!) Chapter 3 - Buoyancy versus gravity = stability (see Chapter Objectives in text) Builds on Chapters 1 and 2 6-week exam is Chapters 1-3!

HYDROSTATICS Review (3.1) Archimedes Principle: – “An object partially or fully submerged in a fluid will experience a resultant vertical force equal in magnitude to the weight of the volume of fluid displaced by the object.” – This force is called the “buoyant force” or the “force of buoyancy”(FB).

HYDROSTATICS Review (3.1) Mathematical Equation: Where... F B is the magnitude of the resultant buoyant force in lb,  is the density of the fluid in lb s 2 / ft 4, g is the magnitude of the acceleration of gravity normally taken to be ft / s 2.  is the volume of fluid displaced by the object in ft 3.

HYDROSTATICS Static Equilibrium : Forces and Moments ( ) Sum of the Resultant Forces: Sum of the Moments about a reference point: Static equilibrium must consist of both conditions!

Hydrostatics The forces lead to translations: Heave Surge Sway The moments lead to rotations: Roll Pitch Yaw Vessel Degrees of Freedom And Static Equilibrium

HYDROSTATICS Static Equilibrium : Stability t B Is this boat in static equilibrium? What are the component forces and moments? Are they internal or external?

HYDROSTATICS Static Equilibrium ( )

HYDROSTATICS Static Equilibrium : Stability (3.2) t B

HYDROSTATICS Changes in the Center of Gravity (3.2) The Center of Gravity (G) is the point at which all of the mass of the ship can be considered to be located (for most problems). It is referenced vertically from the keel of the ship (KG or VCG). (1) Shifting, (2) adding, or (3) removing weight changes the location of the Center of Gravity. (“g” refers to the CG of a weight).

HYDROSTATICS Static Equilibrium : Stability t B Where is the Center of Gravity? The Center of Buoyancy? Are they vertically aligned?

HYDROSTATICS Changes in the Center of Gravity ( ) When weight is added to a ship the G will move in a straight line from its current position toward the center of gravity of the weight being added. G 0 to G f. What happens to the Center of Buoyancy (and the ship)?

HYDROSTATICS Changes in the Center of Gravity ( ) When weight is removed from a ship, G will move in a straight line from its current position away from the center of gravity of the weight being removed. G 0 to G f.

Changes in the Center of Gravity ( ) When a small weight is shifted, G will move parallel to the weight shift but a much smaller distance because it is only a small fraction of the total weight of the ship. HYDROSTATICS

Vertical Shift in the Center of Gravity ( ) Where: KG final is the final vertical position of the center of gravity of the ship as referenced from the keel. KG’s are in “feet”. KG initial is the initial vertical position of the center of gravity of the ship as referenced from the keel.

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) And,  s final is the final displacement of the ship in LT. In this example, it is equal to the initial displacement plus the weight added.  s initial is the initial displacement of the ship in LT. Kg added weight is the vertical position of the center of gravity of the weight being added as referenced from the keel. This line segment is a distance in feet. w added weight is the weight of the weight to be added in LT.

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) The first equation was for a weight addition. What do you do for a weight removal or shift? – Re-examine our first equation:

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) For a weight removal:

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) Similarly, for a weight shift, the final equation is:

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) Combining all three conditions gives a general equation for a vertical change in the Center of Gravity:

HYDROSTATICS Vertical Shift in the Center of Gravity ( ) Last Comments: – The general equation covers all cases for a change in KG. This is the equation you should apply to the exams! See example 3-2 in notes. Examples...

HYDROSTATICS Transverse Shift in the Center of Gravity (3.2.3) Shifts “side to side” of the Center of Gravity. – Starboard is positive and port is negative. As in Vertical case, the Transverse movement of “G” may be caused by either (1) addition, (2) removal, or (3) shifting of weights.

HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) Results in a “List” on the Vessel. – “List” occurs when a vessel is in static equilibrium and down by either the port or starboard side. No external forces are required to maintain this condition and it is permanent unless the Center of Gravity changes. – “List” is different from “heeling”. Heeling occurs because an external couple is acting on the vessel. Heeling is a more temporary condition.

HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) Example (Listing or Heeling?)

HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) From the previous material, which direction does the Center of Gravity move when weight is added, removed or shifted? What happens to the Center of Buoyancy?

HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3)

HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) The Transverse Center of Gravity is referenced in the transverse (athwartships) direction from the centerline of the ship and is labeled TCG. The equation used for a transverse shift in the Center of Gravity is the same as was used for the vertical shift! (With some changes in the notation.)

HYDROSTATICS Transverse Shift in Center of Gravity ( ) Remember a weight shift is just like removing a weight from its original location and adding it to its final location. So for just a weight shift, the generalized equation simplifies to: and:

HYDROSTATICS Vertical and Transverse Changes in “G”

HYDROSTATICS Transverse Shift in Center of Gravity ( ) Review example 3-3 in the text. Note: You can either memorize the entire equations, or be able to derive them from static equilibrium:

HYDROSTATICS Metacenter (3.3) A reference point for hydrostatic calculations for small angles of roll or pitch. Defined as the intersection of the resultant buoyancy forces for a range of roll or pitch between 0 and 10 degrees.

HYDROSTATICS Metacenter (3.3) Beyond 10 degrees roll or pitch, the Metacenter is no longer stationary and becomes less accurate. There is a different Metacenter for ship pitching in the longitudinal direction and ships rolling in the transverse direction. The longitudinal Metacenter is usually times larger.

HYDROSTATICS Metacentric Radius ( ) The distance from the Metacenter to the Center of Buoyancy is defined as the Metacentric Radius (BM)(feet). For small angles on inclination, the Center of Buoyancy moves in a circular arc about the Metacenter.

HYDROSTATICS Metacentric Radius ( )

HYDROSTATICS Metacentric Height ( ) The distance between the Center of Gravity (G) and the Metacenter (M) is defined as the Metacentric Height (GM). – If G is below M, then GM is said to be positive. The ship does not want to capsize. This is GOOD! – If G coincides with M, then GM is said to be zero. A vessel would stay heeled. – If G is above M, the GM is said to be negative. The ship will tip over. This is NOT GOOD!

HYDROSTATICS Metacentric Radius ( ) To determine the values for the previous measurements, we find: KG = KM - GM Where, KM is found from the Curves of Form. GM is found by an inclining experiment or by lengthy, tedious calculations.

HYDROSTATICS Metacentric Radius ( ) KM=KB + BM where: – KB is found by numerical integration – BM is found by: y is the half breadth distance in ft ydx is the area of the differential element on the operating waterplane in ft 2.

HYDROSTATICS Metacentric Radius ( )  s is the submerged volume of the ship’s hull in ft 3 I T is the second moment of the operating waterplane area in the transverse direction with respect to the “x” axis. It has units of ft 4.

HYDROSTATICS Calculating Angle of List (3.4) As a weight shifts across the deck of a vessel, the vessel inclines. How can we predict the angle of inclination? Derivation of Equation – Draw the vessel starting with the Center of Gravity on the centerline so that the vessel will have no list.

Calculating Angle of List (3.4)

HYDROSTATICS Calculating Angle of List (3.4.2) The weight is shifted causing a shift in the Center of Gravity. A moment is created causing the vessel to incline. The underwater shape of the hull changes causing the Center of Buoyancy (B) to move until it is in line with the Center of Gravity (G) and the vessel is back in static equilibrium.

HYDROSTATICS Calculating Angle of List (3.4.3) From the geometry of the situation, we see: Substituting this into the equation for the change in the Center of Gravity we get:

HYDROSTATICS Calculating Angle of List (3.4.3) Where: t is the distance the weight is shifted. This equation only works for small angles because it assumes that the Metacenter does not move.

HYDROSTATICS Inclining Experiment (3.5) Uses small-angle hydrostatics to find the vertical center of gravity (KG) of a ship. Process: – A weight is moved a transverse distance, causing a shift in the TCG, and resulting in measurable inclination (list).

HYDROSTATICS Inclining Experiment (3.5) Navy 44 Incline Experiment

HYDROSTATICS Inclining Experiment (3.5.1) Solving the Angle of Heel equation for the metacentric height, we find: We could measure at one angle of inclination and determine GM, but this would have significant experimental errors, so we measure the inclination with different weights and different positions.

HYDROSTATICS Inclining Experiment (3.5.1) We then plot the data on a graph where the y- axis is the Inclining Moment (wt) and the x- axis is the Tangent of the inclining angle (Tan  ). The average value of GM can be found from the slope of the line. We can see that:

HYDROSTATICS Inclining Experiment (3.5.1) Recall: We want to find the Center of Gravity which can be found by the equation: KM is found from the Curves of Form GM is found from the Inclining Experiment

HYDROSTATICS Inclining Experiment (3.5.2) Removing the Inclining Apparatus we must recalculate KG. This is done as a weight removal problem:

HYDROSTATICS Inclining Experiment (3.5.2) The Inclining Apparatus consists of a long wire with a bob suspended from a tall mast. By measuring the deflection of the wire and knowing the height of the mast, we can calculate Tan .

HYDROSTATICS Inclining Experiment (3.5.3) Shipboard Considerations: – No initial list. – Minimum trim. – Dry bilges. – Liquid fuel and oil to be in accordance with the Shipyard Memo. – Sluice valves closed. – All consumables are to be inventoried. Minimum number of personnel remain onboard. See example 3-4 in your text.

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) A longitudinal shift in the CG will result in the vessel having some trim. Trim is the difference between the forward and aft drafts, T f and T a. It may be calculated by: The Mean Draft is:

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) A vessel is trimmed by the bow when the bow has a deeper draft. This is indicated by a negative trim. A vessel is trimmed by the stern when the stern has a deeper draft. This is indicated by a positive trim. What is the point which the vessel trims about?

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) What happens when a weight is shifted forward or aft? – The vessel goes down by the bow or stern depending on the direction of the weight shift. – Note that the change in trim is independent of the original location of the weight. (i.e. It only matters whether the weight moves forward or aft)

HYDROSTATICS Longitudinal Changes in the Center of Gravity “The Trim Problem” (3.6) Draw a picture of what is happening when a vessel trims due to a weight shift:

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) As the weight shifts forward, a new operating waterline is created and the draft decreases aft and increases forward.

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) We now have two similar triangles and will draw a third which represents the change in trim.

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) To calculate the final drafts we will need: Where we use the similar triangles to find the change in draft due to the weight shift.

HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) Both MT1 and TPI are found from the Curves of Form.