Question

A 85.0 -kg canoe made of thin aluminum has the shape of half of a hollowed-out log with a radius of \(0.475 \mathrm{m}\) and a length of \(3.23 \mathrm{m} .\) (a) When this is placed in the water, what percentage of the volume of the canoe is below the waterline? (b) How much additional mass can be placed in this canoe before it begins to \(\sin \mathrm{k} ?\) (interactive: buoyancy).

Short answer

(a) 7.43% of the volume is below the waterline. (b) 1059 kg additional mass can be placed before sinking.

Step-by-step solution

Calculate the volume of the canoe

The volume of a half-cylinder can be calculated using the formula \( V = \frac{1}{2} \pi r^2 l \), where \( r \) is the radius and \( l \) is the length of the canoe. Substituting the given values, we have: \[ V = \frac{1}{2} \pi (0.475)^2 (3.23) . \] Calculating this gives \( V \approx 1.144 \text{ m}^3 .\)

Determine the buoyant force acting on the canoe

According to Archimedes' principle, the buoyant force \( F_b \) is equal to the weight of the water displaced. The mass of water displaced equals the weight of the canoe for it to float. Hence, \( F_b = mg \), where \( m = 85.0 \text{ kg} \) is the mass of the canoe and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity. \[ F_b = 85.0 \times 9.81 = 833.85 \text{ N} .\]

Calculate the displaced water's volume

The displaced water volume \( V_{displaced} \) is equal to the volume of a mass of water that weighs 833.85 N. The density of water is 1000 \( \text{kg/m}^3\), so the volume can be calculated as follows: \[ V_{displaced} = \frac{F_b}{\text{density} \times g} = \frac{833.85}{1000 \times 9.81} \approx 0.085 \text{ m}^3 .\]

Calculate the percentage of the canoe under the water

The percentage of the volume of the canoe under the water is calculated by dividing the displaced water's volume by the total volume of the canoe: \[ \text{Percentage submerged} = \left( \frac{V_{displaced}}{V} \right) \times 100 = \left( \frac{0.085}{1.144} \right) \times 100 \approx 7.43\% .\]

Calculate the additional mass before sinking

Calculate the total buoyant force needed to submerge the entire canoe using its volume: \[ F_{max} = \rho V g = 1000 \times 1.144 \times 9.81 = 11222.64 \text{ N} .\] Subtract the weight of the canoe to find the max additional mass: \[ m_{additional} = \frac{F_{max} - F_b}{g} = \frac{11222.64 - 833.85}{9.81} \approx 1059 \text{ kg} .\]

Key concepts

Archimedes' Principle

When you think about objects floating or sinking in water, Archimedes' principle plays a crucial role. This principle states that a body submerged in a fluid experiences a buoyant force, which is equal to the weight of the fluid displaced by the body.
This explains why objects float or sink:

• If the object's weight is greater than the buoyant force, it will sink.
• If the object's weight is less, it will float.

In the exercise, the buoyant force acting on the canoe equals the weight of the water displaced. For a canoe to float, the buoyant force must equal the canoe's weight. With Archimedes' principle, we can easily determine how much volume of the canoe is underwater and how much more mass it can carry before it sinks.

Density of Water

The density of an object plays a significant role in determining whether it floats or sinks. In fluid mechanics, the density of water is often a standard reference, typically given as 1000 kg/m³.
When calculating buoyancy, this density value helps determine how much of an object will be submerged.
If an object, like this aluminum canoe, has a lower density than water, it will float. Each time we calculate the volume of displaced water, knowing that the density of water is constant helps make precise and accurate calculations.
When the weight of the water displaced by the canoe equals the canoe's weight, it indicates that equilibrium is achieved, keeping the canoe buoyant.

Volume Calculation

Volume calculation is essential in the context of floating objects. For this canoe, which has the shape of a half-cylinder, the volume is calculated using the formula:
\[ V = \frac{1}{2} \pi r^2 l \]
where \( r \) is the radius, and \( l \) is the length of the canoe. Measuring the radius and length precisely allows us to determine how much space the canoe occupies and therefore how much water it displaces.
In our exercise, after calculating using the given values, we find out that the canoe's whole volume is approximately 1.144 m³. This knowledge is crucial in assessing how much of the canoe stays above or goes below the water when placed in it.

Additional Mass Before Sinking

Once the percent of the canoe submerged is determined, the next interesting finding is how much additional load the canoe can bear before it sinks completely.

With the formula:\[ m_{additional} = \frac{F_{max} - F_b}{g} \]we can see how much extra mass can be safely loaded.
We start by finding the full buoyant force \( F_{max} \) when the entire canoe is submerged, which uses the total volume of the canoe. Next, we subtract the buoyant force when the canoe alone is floating.
The difference gives the force that can support additional weight. Dividing this by the gravitational constant 9.81 m/s² gives the mass in kilograms.
For our canoe, this results in an impressive additional carrying capacity of approximately 1059 kg. This means you can load this canoe with this amount of mass besides its own weight before risking it sinking.