Real World Radical Formulas - Assignment Example

Real World Radical Formulas the al affiliation If one is going to model a ship or boat, the safeness of crew must be a primary aim. One of the most impotent characteristics of boat safeness is the stability. The unique model needs unique investigation of its stability. On the other hand, it is convenient to have some general characteristic to estimate stability of the boat. It is also desirable deal with widely used variables. Thus characteristic is the capsize screening value. To understand the nature of capsize screening value one may consider the rectangle ship model (see figure below).

Now, we assume that the boat deviates from the equilibrium position. The angle of deviation is . It is supposed to be small (see front view on the figure). We put that the origin of coordinate system is in the center of the boat and axis is along the water line. The submerged volume changes on the value for coordinate ( is infinitesimal, is the length of the boat). The corresponding additional momentum value that appears due to the Archimedes force is ( is the water density and is the acceleration of gravity).

The full additional momentum value will be ( is the beam of the boat). According to the Newton’s second law for cyclic motion ( is the moment of inertia). Therefore we have equation for the natural vibrations. The solution is ( is the deviation angle at the initial time moment and is the natural vibration frequency). It is easy to see that. If the natural vibration frequency exceeds some critical value then the boat is unstable. It means that value also must not exceed some limit. To simplify this condition, one can suppose that the boat has some standard form and the centre of the boat is heavier than the edges.

The assumptions give us ( is the mass of the boat and is the some characteristic length). It is well known that, because of the Archimedes law the mass of the boat is equivalent to the displacement of the boat. Finally, we conclude that must not exceed some limit. The cube root of this value multiplied by 4 is called the capsize screening value. The critical value corresponds to feet /pound1/3. Notice, that according to the consideration above, the capsize screening value proportional to the square of the natural frequency.

Problem 103 a. We have:pounds;feetFor the Tartan 4100 capsize screening value is . Therefore we have Problem 103 b. Multiplying the formula by radical of displacement and dividing into for capsize screening value we obtain . The cube of the later formula gives us. The formula defines the displacement value.Problem 103 c. We have:feetSupposing that we have critical value of and using the formula from the problem 103 b one could obtain that Tartan 4100 must be as heavy as pounds to be safe.