Question

In many locations, such as Lake Washington in Seattle, floating bridges are preferable to conventional bridges. Such a bridge can be constructed out of concrete pontoons, which are essentially concrete boxes filled with air, Styrofoam, or another extremely low-density material. Suppose a floating bridge pontoon is constructed out of concrete and Styrofoam, which have densities of \(2200 \mathrm{~kg} / \mathrm{m}^{3}\) and \(50.0 \mathrm{~kg} / \mathrm{m}^{3}\). What must the volume ratio of concrete to Styrofoam be if the pontoon is to float with \(35.0 \%\) of its overall volume above water?

Short answer

The volume ratio of concrete to Styrofoam must be greater than approximately 0.04167 for the pontoon to float with 35% of its overall volume above water.

Step-by-step solution

Find the required buoyancy force for 35% above water

We know that the pontoon will float with 35% of its overall volume above water. This means that 65% of its overall volume must be below water. The buoyancy force must equal the weight of the displaced water for the pontoon to float in equilibrium. If V is the overall volume of the pontoon, then the volume of the water displaced (V_displaced) is: \(V_{displaced} = 0.65 V\) The density of water is approximately \(1000 \mathrm{~kg}/\mathrm{m}^3\). Therefore, the required buoyancy force (F_buoyancy) equals the weight of the displaced water, which is: \(F_{buoyancy}= \rho_{water} \cdot V_{displaced} \cdot g\) where \(g\) is the acceleration due to gravity, approximately \(9.81 \mathrm{~m}/\mathrm{s}^2\). Substituting the expressions for \(V_{displaced}\) and the values of water density and gravity, we get: \(F_{buoyancy} = 1000 \mathrm{~kg}/\mathrm{m}^3 \cdot 0.65 V \cdot 9.81 \mathrm{~m}/\mathrm{s}^2\) \(F_{buoyancy} = 6381.5 V \mathrm{~N}\)

Calculate overall density of the pontoon

The buoyancy force must equal the weight of the pontoon for it to float in equilibrium. The weight of the pontoon (W_pontoon) is: \(W_{pontoon} = m_{concrete} g + m_{Styrofoam} g\) where \(m_{concrete}\) and \(m_{Styrofoam}\) are the masses of the concrete and Styrofoam, respectively. From the definition of density, we have: \(\rho_{concrete} = \frac{m_{concrete}}{V_{concrete}}\) \(\rho_{Styrofoam} = \frac{m_{Styrofoam}}{V_{Styrofoam}}\) So we can express the masses in terms of volume: \(m_{concrete} = \rho_{concrete} \cdot V_{concrete}\) \(m_{Styrofoam} = \rho_{Styrofoam} \cdot V_{Styrofoam}\) Substituting these expressions back into the weight equation: \(W_{pontoon} = \rho_{concrete} \cdot V_{concrete} \cdot g + \rho_{Styrofoam} \cdot V_{Styrofoam} \cdot g\) Setting \(F_{buoyancy} = W_{pontoon}\), we get: \(\rho_{concrete} \cdot V_{concrete} \cdot g + \rho_{Styrofoam} \cdot V_{Styrofoam} \cdot g = 6381.5 V\) Now, in order for the pontoon to float, the overall density, \(\rho_{overall}\) must be less than the water density: \(\rho_{overall} = \frac{\rho_{concrete} \cdot V_{concrete} + \rho_{Styrofoam} \cdot V_{Styrofoam}}{V} < 1000 \mathrm{~kg}/\mathrm{m}^3\)

Calculate the volume ratio of concrete to Styrofoam

Let's represent the volume ratio of concrete to Styrofoam as R, where: \(R = \frac{V_{concrete}}{V_{Styrofoam}}\) In order to calculate R, we need to express \(V_{concrete}\) and \(V_{Styrofoam}\) in terms of the total volume V. We have: \(V = V_{concrete} + V_{Styrofoam}\) and \(R = \frac{V_{concrete}}{V_{Styrofoam}}\) First, solve for \(V_{Styrofoam}\) from the volume ratio expression: \(V_{Styrofoam} = \frac{V_{concrete}}{R}\) Now, substitute this into the total volume expression: \(V = V_{concrete} + \frac{V_{concrete}}{R}\) Next, let's set up an expression for the overall density in terms of the volume ratio R, using the total volume and overall density expressions. We know that \(\rho_{overall} < 1000 \mathrm{~kg}/\mathrm{m}^3\), so: \(\frac{\rho_{concrete} \cdot V_{concrete} + \rho_{Styrofoam} \cdot \frac{V_{concrete}}{R}}{V} < 1000 \mathrm{~kg}/\mathrm{m}^3\) Finally, we will solve for the volume ratio R, given the densities of concrete and Styrofoam: \(\frac{2200 \mathrm{~kg}/\mathrm{m}^3 \cdot V_{concrete} + 50 \mathrm{~kg}/\mathrm{m}^3 \cdot \frac{V_{concrete}}{R}}{V} < 1000 \mathrm{~kg}/\mathrm{m}^3\) Multiply both sides by V: \(2200 \mathrm{~kg}/\mathrm{m}^3 \cdot V_{concrete} + 50 \mathrm{~kg}/\mathrm{m}^3 \cdot \frac{V_{concrete}}{R} < 1000 \mathrm{~kg}/\mathrm{m}^3 \cdot V\) Rearrange and solve for R: \(R > \frac{50 \mathrm{~kg}/\mathrm{m}^3 \cdot V_{concrete}}{(1000 \mathrm{~kg}/\mathrm{m}^3 - 2200 \mathrm{~kg}/\mathrm{m}^3) \cdot V}\) Simplifying: \(R > \frac{50 \mathrm{~kg}/\mathrm{m}^3}{-1200 \mathrm{~kg}/\mathrm{m}^3}\) Calculating the volume ratio R: \(R > 0.04167\) So the volume ratio of concrete to Styrofoam must be greater than approximately 0.04167 for the pontoon to float with 35% of its overall volume above water.

Key concepts

Buoyancy Force Calculation

Understanding how to calculate buoyancy force is crucial when designing structures like floating bridges. Buoyancy is a force exerted by a fluid that opposes an object's weight. In our scenario with a floating bridge pontoon, this force must balance the pontoon's weight for it to stay afloat.

By Archimedes' principle, the buoyancy force equals the weight of the fluid that the submerged part of the pontoon displaces. As the pontoon floats with 65% submerged, only that volume of water contributes to buoyancy. Thus, the formula for buoyancy force involves multiplying the density of the water (\(1000 \frac{kg}{m^3}\)), the volume submerged (\(0.65V\)), and the acceleration due to gravity (\(9.81 \frac{m}{s^2}\)).

Density and Flotation

Density plays a pivotal role in flotation. A floating object must have an overall density less than the fluid it's floating in. The densities of the pontoon's materials—concrete and Styrofoam—determine whether it sinks or floats.

In this exercise, concrete has a much higher density than water, while Styrofoam is significantly lighter. The combination of the two materials achieves an average density that allows the pontoon to float. The average density, for flotation purposes, is calculated by adding the mass of concrete and Styrofoam and dividing by the pontoon's total volume. Hence, to float with a specific portion above water, the ratio of concrete to Styrofoam must be carefully adjusted.

Volume Ratio Determination

The volume ratio between concrete and Styrofoam determines the pontoon’s ability to float and impacts how much of it remains above the waterline. To establish a stable floating bridge, the correct balance must be calculated.

In this case, we're seeking a volume ratio that allows the pontoon to keep 35% of its volume above water. To find the proper ratio, we first express each material's volume as a fraction of the total volume. Since the overall density needs to be less than that of water, we set up an inequality that incorporates both materials' densities and solve for the volume ratio. The outcome ensures the average density is below water’s density, meaning the adjust ratio is essential for the pontoon to have buoyancy with the desired proportion above the surface.