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ENGR101 Assignment 3 ENGR101 Foundations of Engineering

Engineering Process for the Design and Build of a Model Submarine


A submarine model was to be built at a cost of <$20 with recycled materials that could move horizontally fully submerged at least for a distance four times its body length. This was tested using a twisted rubberband powered submarine with calculated mass required to achieve neutral buoyancy.

The submarine needed a mass of 742.3 grams in steel to maintain stability and neutral buoyancy which was 9 grams more than that estimated. The submarine using the thicker rubberband travelled an average distance of 137 cm which was greater than 4 times the submarine’s length of 23 cm, compared to that with the thin rubberband’s distance of 84 cm. Hence, the submarine model satisfied all constraints.

In a separate experiment on water density versus buoyancy, it was found decreasing density decreased buoyancy. A trial on decreasing buoyancy versus increasing floating time showed the float time increased substantially close to neutral buoyancy.


A submarine model that stays fully submerged and level and is able to move horizontally for a distance of four times its length is to be constructed. In order for the model to be successful, a few constraints need to be set in place.

Hard constraints:

● Must be able to stay fully submerged under water and level

● Should be able to move horizontally for a distance equivalent of at least 4 times its body length

● The materials used to construct the submarine must not exceed the budget of 20 NZD

Soft constraints:

● Easy to construct

● Uses recycled materials to save costs


Four forces act on submarines while they are submerged in water. These are:

● Buoyancy – the force acting on the object to make it float

● Engine Force – or propelling force, acts on the object to make it move

● Water resistance – the force which decelerates the object as it moves through the water (opposes the engine force)

● Weight – the force that pulls the submarine down due to gravity (opposes buoyancy)

For submarines to be fully submerged and level, but not fully sunken, it must maintain neutral buoyancy. At neutral buoyancy, the buoyancy and weight of the submarine are equal. Buoyancy can be calculated as follows:

Buoyancy (Newtons) = V (cm3)× G (cm/s−2)× ρ (g/cm3 ) ÷1000

Where V is volume of water displaced, G is gravity (980 cm/s−2 ) and ρ​ is water density.​

Theoretically, the constructed submarine material weight minus buoyancy force equivalent must provide the mass of material required to be placed in the submarine to achieve neutral buoyancy. However, in practice there are errors in volume estimates owing to density assumptions and using the conventional water density value of 1 g/cm​3​, which could be normally less because the above density is achieved at 4​o​C.

As for the propelling force in the submarine model was not to come from pushing the submarine manually, so the force needs to be generated continuously from some sort of motor or mechanical force. Hence a viable solution to this problem would be a simple rubberband powered submarine. The rubberband would only need to be wound at the beginning with sufficient torque to move the submarine without any other human pushing force. The rubberband driven submarine is easy to build and the materials would cost less than $20, to satisfy the constraints.

Calculating Buoyant Force as a function of Water Density

In a separate experiment, the buoyancy was measured as a function of water density (Table 1). The densities were obtained at different water temperatures (Shapley, 2011). This was because water temperature can change the density of the water which in turn can affect the buoyancy of the submarine.

Sample Calculation for Buoyancy with water density of 0.999972:

V olumeof submarine × Gravity = constant

784.8 × 980 = 769104

Buoyancy = V × G × ρ

Buoyancy = Constant × ρ


= 769.082465 N

Table 1: Data for buoyancy as a function of water density

Figure 1: Buoyancy as a function of water density

As seen in Figure 1, decreasing water density resulted in decreasing buoyancy. Even a small decrease in density can cause large reduction in buoyancy. For example, approximately, with 0.008 g/c 3​ reduction in water density can reduce buoyancy by 6 N. Conversely, increasing water density can result in increased buoyancy. Consequently, a submarine will have greater buoyancy in sea water compared to freshwater.

For an object to be suspended in water, the object must be neutrally buoyant. This means that the weight of the object equals buoyancy force (OpenStaxCollege, 2012) (1). (g is cancelled out in both sides of the equation)

i.e massof submarine + mass of contentsof submarine = volume ×density (1)

Experimental Methods

Materials (all costs reported in NZD): 2 Paper clips - $0.06, 3 rubberbands - $0.30, plastic bottle, container lid, 2 popsicle sticks - recycled. Total cost on build equals $0.36 cents.

To construct the submarine model (KiwiCo, 2017) in Figure 1, two holes were punctured on the bottle - one on the centre of the bottom of the bottle and one on the centre of the bottle cap. This allowed both paper clips to feed through the holes and into the bottle. The clip wire protruding outside the bottom of the bottle was bent in U shape and inserted into a hole punched adjacent to the bottle bottom centre hole to enable clip being securely attached to the bottle bottom.

A propeller was cut out from the yoghurt container lid, each blade was twisted to provide swirling feature, a hole was punched in the centre and the protruding paper clip wire from the bottle cap end was fed through the hole. Another small hole was punched on the propeller adjacent to the centre of the propeller and the protruding cap end wire was bent in U shape to feed through the new hole to make contact with the bottle cap to prevent propeller from slipping on the wire during propulsion.

The ends of the paperclips inside the bottle were bent in U shape using a pair of chopsticks, and the rubber-band was then hooked onto the bottle bottom end clip. A used wire coat hanger was converted into a small U shape hook and the other end of the rubberband was pulled with tension and hooked onto the cap end of clip wire hook. Two popsicle sticks with 10 cent coins on each end were attached to the bottom of the bottle with rubberbands, to act as the submarine’s fins as shown in Figure 1.

Figure 1: The constructed submarine model with a length of 23 cm.

The submarine model was tested in a spa bathtub (Figure 2) where no other objects could affect the path of the submarine.

Figure 2: Testing environment with a length of 150 cm.

The distance travelled underwater of the submarine was measured from when the submarine began propelling until it fully stopped. The distances were measured 3 times with the submarine propelled by rubberbands with two different thicknesses. The average of the 3 distances was calculated.

Results and Discussion

Achieving neutral buoyancy trial

The equation from (1) can be utilized to find the mass of the contents of the submarine which will make the submarine neutrally buoyant, where the mass of the empty submarine is 51.5 grams and the volume is 784.8 grams (see volume calculations in Appendix 1).

massof contents of submarine = 784.8 − 51.5

= 733.3 grams

That is, by calculation, 733.3 g of mass is required to make the submarine neutrally buoyant. The first mass trialled to submerge the submarine was water. However, owing to water bubble (water space in the submarine) being unstable in the bottle, the submarine could not be stabilised horizontally at neutral buoyancy. Subsequently, glass marbles were used as weights, however, it was found owing to the density of

2.57 g/cm​3​, the volume occupied by the marbles (285 cm​3​) began to interfere with rubberband mechanism. Consequently, it was decided to use steel materials with greater density of 8.05 g/cm​3​ providing lower volume of added weights. Since considerable accuracy in mass used was required a pocket digital balance with 0.01 gram mass graduation was used for the above purpose.

The submarine was weighted with 733.3 g of steel nails and bolts and tested for buoyancy. The submarine was afloat at the above added weight hence required more weight indicating the calculated mass required was an underestimation. It was found by adding additional mass to achieve neutral buoyancy 742.3 g of nails and bolts were required, indicating a difference of 9.0 g between calculated and actual mass required. This demonstrates that at or close to neutral buoyancy, even small weight difference can make a substantial effect on the submarine either floating or sinking.

Designing propelling force for the submarine

It was decided to utilise twisted rubberband torque to propel the submarine. This was because by twisting the rubberband, elastic potential energy is stored in it, which would then turn into rotational kinetic energy once the rubberband starts unwinding. According to test done by Allain (2018), a thick rubberband produced an energy storage of 97.1 Joules at 100 revolutions, compared to a thin rubberband that produced 7.2 Joules. Moreover, in the same trial, by twisting the thick rubberband, more torque was produced per rotation in comparison to the thin rubber band (4.92​ x 10​-4​ N*m ​ versus ​ 3.68​ x 10​-5 ​N*m). ​

The submarine was first trialled with a thin, long rubberband as the motor, and then a thick, long rubberband. The properties of the two rubberbands used in the experiment is provided in Table 2 and the submarine distance trial is provided in Table 3.

Table 2: Properties of rubberbands.

Thin Rubberband

Thick Rubberband


1 mm

5 mm

Length without stretch

6.2 mm

10.5 mm

Length stretched

21.5 mm

40 mm


35 times

25 times

Table 3: Distance the submarine travelled underwater with different rubberband thicknesses.


ce travelled underwater (cm)

Trial 1

Trial 2

Trial 3

Average Distance

Thin rubberband





Thick rubberband





Using a thin rubberband, the submarine’s propeller had to be wound much more than when using the thick rubberband. This was because the thin rubberband did not produce as much torque to apply on the propeller. The thin rubberband also was fragile and could have chances of fraying and snapping if wound excessively.

The thick rubberband was stronger and therefore able to produce more torque and apply it to the propeller. As more torque is applied to the propeller, more force is pushed onto the propeller from the rubberband, resulting in the submarine moving further. As shown by the results in the table, the thick rubberband made the submarine propel further than the thin rubberband, with an average distance of 137 cm.

Part of the success criteria for the submarine model is that it should be able to move horizontally for a distance equivalent of 4 time its body length. The submarine model constructed was 23 cm long, which means it needs to move 92 cm in order to be successful.

Time for Submarine Model to Reach Surface

The time for the submarine model to rise to the surface, with several different masses was also measured and compared to the calculated theoretical values (Xodin, 2012) (Table 3). As the mass added on to the submarine increased, the time for the submarine to reach the water surface also increased. This was because, in relation to the equation Fb = ρ ×V ×g , ρ×V = M which means Fb = M ×g. Therefore, if the mass of the submarine increases, the buoyancy force caused by the weight of the displaced water is countered hence the travel time will increase owing to reducing buoyancy. This is also illustrated in Appendix 2 calculations where, t is inversely related to resultant buoyancy force where decreasing buoyancy force means increasing travel time for a given distance travelled.

A large chilly bin was used to trial the buoyancy versus travel time taken for the submarine released from the bottom of the chilly bin to the surface of the water. The depth of water which was distance travelled by the submarine was maintained at 27 cm throughout the trial. An accurate cell phone stop watch was used to measure the time taken for the submarine to reach the surface of water from the bottom of the chilly bin.

Glass marbles were used to increase the mass and decrease the buoyancy force by adding known mass gradually by measuring by using a digital pocket scale with 0.01 gram graduation. The time taken to travel 27 cm distance for each additional mass was compared against the calculated time as per Appendix 2.

It was found as additional mass was added, the time taken travel the distance increased. Once reached a mass of 756.33, one glass marble (5 grams) was added each time since the increase in travel time was more pronounced with small addition of mass.

Once the travel time reached 8.96 seconds, any addition of the weight (around 5 grams) resulted in the submarine sinking which meant there was negative buoyancy owing to greater weight than the buoyancy provided by the submarine. At the above travel time, the total weight of the submarine was 783.98 grams hence neutral buoyancy would have required a fraction of more weight.

As seen in Table 3, the calculated travel time was lower than the measured time. This may have been caused by errors in calculating travel time which must have bene caused by density assumptions and volume estimates of submarine materials.

Table 3: Measured and calculated times for submarine to rise to surface, with different masses (additional to submarine mass)


The hard constraints the solution had to satisfy were the submarine had to stay submerged and level underwater, move a distance equivalent to four times its length and construction had to cost less than $20.

The proposed solutions were, 742.3 grams in steel to be added into the submarine body to achieve stability and neutral buoyancy with an unimpeded twisted thick rubberband powered submarine which achieved all objectives and overcame constraints. While fully functional, the thin rubberband powered submarine failed to propel the submarine above the required travel distance because of lack of torque.

It was also demonstrated that buoyancy can be sensitive to factors affecting buoyancy such as water density, material volume (volume of water displaced) and weight and water temperature. It was also found that closer to achieving neutral buoyancy a minor difference in buoyancy versus weight of the submarine can make it to float or sink. The engineering process was successfully used in this investigation to design, build, test and evaluate the various solutions proposed in order to find a model that best fulfilled the definition of success.


KiwiCo (2017). ​Submarine​. Retrieved from:

Shapley, P. (2011). ​Temperature Effects on Density​. University of Illinois. Retrieved from:

OpenStaxCollege. (2012) ​Archimedes’ Principle.​ College Physics. Retrieved from:

Allain, R. (2018). ​How much Energy can you store in a Rubber Band​. Retrieved from:

Xodin. (2012) ​Time it takes for submerged object to rise to surface​. Physics Forums. Retrieved from: 681/

Vitz, E., Moore, J., Shorb, J., Prat-Resina, X., Wendorff, T., Hahn, A. (2019) ​Sorting Recyclable Plastics by Density.​ LibreTexts. Retrieved from: ental_and_Green_chemistry/Sorting_Recyclable_Plastics_by_Density

Answers. (2008) ​What is the density of a popsicle stick? ​Retrieved from:

AmesWeb. (2018) Density of steel. ​ ​Retrieved from


Appendix A

This appendix shows the calculation for finding the required mass needed for

Buoyancy of the submarine model = volume of water displaced V x gravity g x water density d

Volume of water displaced V = volume of the plastic bottle + paddles + coins (volume of the propeller was considered as negligible)

Volume of plastic bottle = water fillable space in the bottle (754.5 cm ​3​) + volume of plastic material (25.8 cm​3​) ( mass of the bottle 34.8 cm ​3 ÷ density of the bottle 1.35 g/cm ​3​) = 780.3 cm​3 ​(density of clear plastic from Vitz et al., 2019)

Paddles volume = mass (2.18 g) ÷ density (0.75 g/cm​3​) = 2.9 cm​3 ​( density of wood from Answers, 2008)

Coins volume = mass (13.1 g) ÷ density (8.05 g/cm​3​) = 1.6 cm​3​ (density of steel from AmesWeb, 2018)

Therefore V = 780.3 + 2.9 + 1.6 cm​3​ = 784.8 cm​3

Buoyancy of the submarine = 784.8 cm​3​ x 980 g/s​-2​ x 1 g/cm​3​ ÷ 1000 = 769.1 N

The total mass of the unweighted submarine = mass of bottle (34.8 g) + mass of paddles (2.2 g) + mass of coins (13.1 g) + rubberband (1.4 g) (2 paper clip mass was considered negligible) = 51.5 g

Appendix B

This appendix shows a sample calculation for calculating the time taken for the submerged submarine model to reach the surface.

The calculated time for the submarine to rise to the surface was calculated by using the linear mechanics equation:

t = √ aL

Where L is the depth of the bin used, 27 cm, a is the acceleration, and t is time.

The net force (upward force - downward force) of the submarine had to be first calculated to calculate acceleration.

F = buoyancy forceweight

F = ρ×V ×g mg

= 1 × 784.8 ×980 − 634.8 ×980

= 147000 a = F/m a = 147000/634.8 a = 231.56


= 0.482 seconds

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