Problem 68 - University Physics with Modern Physics

Question

When an open-faced boat has a mass of 5750 kg, including its cargo and passengers, it floats with the water just up to the top of its gunwales (sides) on a freshwater lake. (a) What is the volume of this boat? (b) The captain decides that it is too dangerous to float with his boat on the verge of sinking, so he decides to throw some cargo overboard so that 20\(\%\) of the boat's volume will be above water. How much mass should he throw out?

Step-by-Step Solution

Answer

(a) 5.75 m³ (b) Throw out 1150 kg of mass.

Step 1: Understand the Problem and Identify Formula

We need to use the principle of buoyancy, which states that the buoyant force equals the weight of the displaced fluid. Here, the weight of the boat equals the weight of the displaced water when the boat is on the verge of sinking.

Step 2: Calculate Volume of the Boat

For part (a), we use the fact that the boat displaces a volume of water equal to its mass. The density of freshwater is approximately 1000 kg/m³. This means the volume \( V \) of the boat equals:\[ V = \frac{\text{mass of the boat}}{\text{density of water}} = \frac{5750 \text{ kg}}{1000 \text{ kg/m}^{3}} = 5.75 \text{ m}^{3} \]

Step 3: Determine Volume Above Water

For part (b), the captain wants 20% of the boat's volume above water. We calculate this:\[ V_{\text{above}} = 0.20 \times 5.75 \text{ m}^{3} = 1.15 \text{ m}^{3} \]

Step 4: Determine Volume Submerged

Calculate the new submerged volume, which is 80% of the boat's total volume:\[ V_{\text{submerged}} = (1 - 0.20) \times 5.75 \text{ m}^{3} = 0.80 \times 5.75 \text{ m}^{3} = 4.60 \text{ m}^{3} \]

Step 5: Calculate New Mass and Determine Cargo to Remove

The submerged volume relates to the new total mass of the boat (including passengers and cargo). With the new submerged volume:\[ \text{New mass} = V_{\text{submerged}} \times \text{density of water} = 4.60 \text{ m}^{3} \times 1000 \text{ kg/m}^{3} = 4600 \text{ kg} \]To find out how much mass should be thrown overboard:\[ \text{Mass to throw out} = 5750 \text{ kg} - 4600 \text{ kg} = 1150 \text{ kg} \]

Key Concepts

Density of Freshwater

The concept of density is crucial when dealing with buoyancy and floating objects. Density refers to how much mass is contained within a specific volume. It is usually expressed in units of kilograms per cubic meter (kg/m³). For freshwater, the density is typically about 1000 kg/m³, a standard value you will often use in calculations involving water.
The density of freshwater creates the buoyant force that helps objects to float. It determines how much water needs to be displaced for an object to stay afloat. In our exercise about the boat, understanding this density allows us to calculate how much volume of water the boat displaces.
By using the formula \[ V = \frac{\text{mass of the object}}{\text{density of water}} \] you can find the volume of water displaced, which is crucial to understanding buoyancy.

Buoyant Force

Buoyant force is a natural force exerted by a fluid (like water) that opposes the weight of an object immersed in it. It is the reason why objects float. When you drop an item in water, it pushes aside a certain amount of water equal to its own volume.
According to the principle of buoyancy, the buoyant force on an object is equal to the weight of the fluid it displaces. This means for our floating boat example, the boat's weight is balanced exactly by the weight of the water it displaces.
If an object floats, it means the buoyant force equals the object's weight, and thus it does not sink. For a boat buoyed up to its gunwales, as the exercise suggests, it is on the verge of having its weight exceed the buoyant force, which would cause it to sink.

Mass and Volume Calculations

Calculating mass and volume plays a crucial role in understanding buoyancy and displacement. Starting with the mass of an object, this is a measure of the amount of matter in the object and is typically in kilograms (kg).
Volume, on the other hand, is the amount of space an object occupies, measured in cubic meters (m³). By knowing both the mass and density of a material, you can calculate the volume using: \[ V = \frac{\text{mass}}{\text{density}} \]
In our exercise, the mass of the boat was given as 5750 kg, and the density of the water was taken as 1000 kg/m³. The initial calculation resulted in the boat needing to displace a volume of 5.75 m³ of water to float without sinking.
The follow-up question required us to assess how reducing the boat's weight alters its buoyancy, ensuring that 20% of the boat remains above water, solving for the volume of 1.15 m³ that remains unsubmerged and calculating the necessary weight to offload.