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A \(62.0-\mathrm{kg}\) survivor of a cruise line disaster rests atop a block of Styrofoam insulation, using it as a raft. The Styrofoam has dimensions \(2.00 \mathrm{~m} \times 2.00 \mathrm{~m} \times 0.0900 \mathrm{~m}\). The bottom \(0.024 \mathrm{~m}\) of the raft is submerged. (a) Draw a force diagram of the system consisting of the survivor and raft. (b) Write Newton's second law for the system in one dimension, using \(B\) for buoyancy, \(w\) for the weight of the survivor, and \(w_{r}\) for the weight of the raft. (Set \(a=0\).) (c) Calculate the numeric value for the buoyancy, \(B\). (Seawater has density \(1025 \mathrm{~kg} / \mathrm{m}^{3}\).) (d) Using the value of \(B\) and the weight \(w\) of the survivor, calculate the weight \(w_{r}\) of the Styrofoam. (e) What is the density of the Styrofoam? (f) What is the maximum buoyant force, corresponding to the raft being submerged up to its top surface? (g) What total mass of survivors can the raft support?

Step by step solution

01

Draw Force Diagram

In the force diagram, there are two forces acting downwards namely the weight of the survivor (\(w\)) and the weight of the raft (\(w_r\)), and one force acting upwards called the buoyant force (\(B\)). These forces are in equilibrium as the system is at rest.

02

Apply Newton's Second Law

According to Newton's Second Law, the net force acting on an object at rest is zero. Hence, \(B = w + w_r\).

03

Calculate Buoyant Force

The buoyant force, \(B\), is equal to the weight of the sea water displaced. Hence, \(B = V蟻_{water}g\) where \(V\) is the volume submerged, \(蟻_{water}\) is the density of water and \(g\) is the acceleration due to gravity. Given volume \(V = 2.00 m \times 2.00 m \times 0.024 m\), density of water \(蟻_{water} = 1025 kg/m^3\) and \(g = 9.8 ms^-2\), solve for \(B\).

04

Calculate weight of Raft

The weight of the raft, \(w_r\), can be obtained from the equation in Step 2, \(w_r = B - w\), where \(w\) is the weight of the survivor (\(w = mg\), \(m = 62.0 kg\), \(g = 9.8 ms^-2\)). Using the calculated values solve for \(w_r\).

05

Calculate Density of Styrofoam

The density of the Styrofoam can be obtained by \(蟻 = \frac{w_r}{V_r g}\), where \(V_r\) is the total volume of the Styrofoam (\(V_r = 2.00 m \times 2.00 m \times 0.0900 m\)). Using the the calculated value of \(w_r\) from step 4, solve the equation to get \(蟻\).

06

Calculate Maximum Buoyant Force

The maximum buoyant force is obtained when the raft is completely submerged, i.e, the volume of displaced seawater is equal to the total volume of the raft as defined in Step 5. Then \(B_{max} = V_r蟻_{water}g\).

07

Calculate Total Mass of Survivors

The total mass of survivors the raft can support, \(m_{total}\), can be found when the total weight of the survivors and the raft equals the maximum buoyant force calculated in step 6. Therefore, \(m_{total} = \frac{(B_{max}-w_r)}{g}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Force Diagram

When considering a physical problem involving buoyancy and forces, it is helpful to start by creating a force diagram. A force diagram, or free-body diagram, visually represents all the forces acting upon an object. In this scenario, we have two main components: the survivor and the Styrofoam raft.

In the force diagram:

• The weight of the survivor (\(w\)) acts downward.
• The weight of the raft (\(w_r\)) also acts downward.
• The buoyant force (\(B\)) acts upward, countering the weights.

Since the system is in equilibrium (neither sinking nor rising), the upward buoyant force is precisely balanced by the combined downward forces of the survivor and raft. This visual tool is crucial for understanding how forces are distributed in the system.

Newton's Second Law

Newton's Second Law, fundamental in physics, states that the sum of the forces acting on an object results in an acceleration of that object. However, when an object or system is in equilibrium鈥攁s in this exercise鈥攖he net force is zero, and there is no acceleration.

The equation we derive is:\[ B = w + w_r \] where:

• \( B \) is the buoyant force,
• \( w \) is the weight of the survivor, and
• \( w_r \) is the weight of the raft.

By setting the acceleration (\( a = 0 \)), we validate that the upward buoyant force equals the sum of the downward forces. This simple yet powerful law allows us to understand balanced forces and how they affect the motion or stability of bodies.

Density

Density is a physical property that indicates how much mass is contained in a given volume. It is mathematically expressed as:\[ \rho = \frac{m}{V} \] where \( \rho \) is density, \( m \) is mass, and \( V \) is volume.

In this problem, we use density to find the characteristics of both the seawater and the Styrofoam. The seawater has a known density of \( 1025 \, \text{kg/m}^3 \). For the Styrofoam, once we have determined its volume and weight (\( w_r \)), we rearrange the formula to\[ \rho_{\text{styrofoam}} = \frac{w_r}{V_r \, g} \] This calculation provides the material's density, which helps us understand buoyancy鈥攐bjects less dense than water float.

Buoyant Force Calculation

Buoyancy describes the force exerted by a fluid that opposes an object's weight. It is the reason why objects float or sink in water. Archimedes' principle tells us that the buoyant force is equal to the weight of the fluid displaced by the submerged part of the object.

For calculation:\[ B = V \rho_{\text{water}} g \] where:

• \( B \) is the buoyant force.
• \( V \) is the volume of seawater displaced (submerged volume of the raft).
• \( \rho_{\text{water}} \) is the density of seawater.
• \( g \) is the acceleration due to gravity (\( 9.8 \, \text{ms}^{-2} \)).

By using this formula, you can compute how much force is keeping the raft afloat. For maximum buoyant force, we consider the entire Styrofoam volume submerged.