Math and Science Report

The ROV Mate Submersible is a complicated project that entails many aspects of science, math and technology. Incorporated particularly in the frame of the ROV; the science and technology are the two predominating factors for success; math plays a major role, involving buoyancy and the angles that the pieces are put together at.
Science and Technology:

Zip tie:

Zip ties are a hard plastic, reusable strap that can hold two objects very tightly together. They are cheap and easy to come by and good for many situations. They will be used to strap the plastic mesh that will hold all of the components onto the PVC frame.

The Chemistry of PVC:

Polyvinyl chloride, better known as PVC, it is a thermoplastic (see figure 1 below for chemical make up). A thermoplastic is a plastic that can be melted into a liquid and put into molds to make the desired piece. It is generally used in piping systems in commercial structures in today’s modern world (Wikipedia). This material is durable, cheap, easy to come by and very customizable in the fact that an endless number of structures can be built with it according to the situation in which it best fits.

(Figure 1) – The chemical make up of polyvinyl chloride on the right hand of the equation. It is made up of hydrogen (H), chlorine (Cl), and carbon (C). This than goes through polymerization and becomes what is PVC on the left (Wikipedia).

PVC Cement:

PVC cement is a critical tool in putting together several lengths and/or joints of PVC. It chemically bonds the PVC to create a nearly unbreakable, airtight seal when applied correctly. The chemicals in PVC could not have been combined many years ago, but due to today’s technology this is possible. PVC cement contains: acetone, methyl ethyl ketone, cyclohexanone, tetrahydrofuran, and finally polyvinyl chloride. All of these chemicals are dangerous to humans if ingested but together when mixed properly they make an adhesive for plastics that is strong as cement.
Mathematics:
Buoyancy:

Buoyancy is the waters force pushing upward on an object, in this specific case the ROV Mate Frame. To calculate the buoyancy of the craft, an equation for the volume will be necessary. The way to most accurately measure the volume of the craft will be to measure all of its components as cylinders. This will give a relatively accurate volume of the craft. The equation for the volume of a cylinder is as follows: V= 3.14r²h

The “r” or radius for all of the piping in the frame is 1in because the diameter is 2 in. The “h” or height for the two long sides seen in Figure 3 will have a height of 31.2 in. The shorter sections will have a height of 13 in. And the six short bottom pieces will have a height of 2.5 in.

V= 3.14 (1²) 31.2 in = 97.97 in³ x 2 = 195.94 in³

V= 3.14 (1²) 13 in = 40.82 in³ x 2 = 81.64 in³

V= 3.14 (1²) 2.5 in = 7.85 in³ x 6 = 47.1 in³

Total volume = 195.94 in³ + 81.64 in³ + 47.1 in³ = 324.68i n³

Conversion from cubic inches to liters of displacement:

324.68 in³ x 0.01639 = 5.322 liters

The crafts overall buoyancy and displacement of water is roughly 324.68 or 5.322 liters.

Physical Check:

To check the math, the crafts underwater submerged volume will be needed. This measurement will be obtained after the frame is built. The method to check the crafts volume will be based on Archimedes Principle; when an object is placed in a liquid the amount of fluid that it displaces is its . The craft will then be submerged in a measured body of water. After the craft is placed in the water the displacement will be measured by the change in height of the water. This displacement is the submerged volume. After finding the submerged volume this measure will be multiplied with the density of the surrounding liquid. This liquid will be water. The density of water is 1gram/cm³. The ROV will not be operated at a depth of more than four meters so the pressure will be approximately one atmosphere, or G= 9.81. The equation that will be used to check the math:

Buoyancy = (1gram/cm³) (Volume) (9.81)

After the calculation is made weights will be added evenly to the “F” pieces displayed in Figure two below. The weights will be added until the craft is close to neutral buoyancy yet is still positively buoyant.

Pythagoras Theorem:

A² + B² = C². This simple equation can be used to measure the length of any side of a right angle triangle, if the other two dimensions are known. This will help the construction of the frame so that the corner angles on the top rectangle (see Figure 2; the combined pieces of “A," "B," "D," and "E"). When "A," "B," and "D" are combined the angle must be precisely at 90˚. In order to do this the lengths of two adjacent sides must be taken (see Figure 3). The longer side length of the top of the frame is 2’-6”. The shorter side length is 1’-6”. These are sides A and B.

A = 1.5’

B = 2.5’

A² = 2.25’

B² = 6.25’

2.25’ + 6.25’ = 8.5’ the square root of 8.5’ is 2.915’. When the frame is being dry fitted together the distance from corner to corner must be 2’-11”.

These sciences, mathematics, and technology all contribute to the soundness of the design and the final production of the frame. The mathematics will help make the craft the perfect dimensions. The calculation of buoyancy will aid in the counter weighting the crafts positive buoyancy even after all of the components are added.
Pictures:

(Figure 2) – The completed expanded isometric CAD drawing of the ROV Mate Submersible frame.

(Figure 3) - This is the completed labeled orthographic view of the ROV Mate frame.