Question

Archimedes' principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let \(\rho_{w}=1 \mathrm{g} / \mathrm{cm}^{3}=1000 \mathrm{kg} / \mathrm{m}^{3}\) be the density of water and let \(\rho\) be the density of an object in water. Let \(f=\rho / \rho_{w}\). If \(01,\) then the object sinks. Consider a cubical box with sides 2 m long floating in water with one-half of its volume submerged \(\left(\rho=\rho_{w} / 2\right) .\) Find the force required to fully submerge the box (so its top surface is at the water level).

Short answer

Answer: 0 N

Step-by-step solution

Calculate the weight of the floating box.

First, since the fraction submerged is 0.5, we can write the buoyant force as the weight of the water displaced: \(F_B = V_\text{submerged} \rho_w g\), where V_submerged is the submerged volume, \(\rho_w\) is the water density, and g is the acceleration due to gravity. The submerged volume (half-volume, since 1/2 of the cube is underwater) is: \(V_\text{submerged} = (2\,\text{m})^3 \times \frac{1}{2} = 4\,\text{m}^3\). Thus, the buoyant force is: \(F_B = 4\,\text{m}^3 \times 1000\,\text{kg} / \text{m}^3 × 9.81\,\text{m} / \text{s}^2=39240\,\text{N}\). #Step 2 - Weight of the fully submerged box#

Calculate the weight of the fully submerged box.

Now, we calculate the weight of the fully submerged box (when the top surface is at the water level). The volume of the box is: \(V_\text{box} = (2\,\text{m})^3 = 8\,\text{m}^3\). The mass of the box is given by: \(m_\text{box} = \rho V_\text{box} = (\rho_w / 2) \times 8\,\text{m}^3 = 4000\,\text{kg}\). The weight of the box is: \(W_\text{box} = m_\text{box} g = 4000\,\text{kg} \times 9.81\,\text{m} / \text{s}^2 = 39240\,\text{N}\). #Step 3 - Force required to fully submerge the box#

Calculate the force needed to submerge the box.

To fully submerge the box, we need to overcome the buoyant force. Since the weight of the floating box equals the buoyant force and the weight of the fully submerged box is equal to the weight of the floating box, it follows that the force required to fully submerge the box is simply the difference between these forces. \(F_\text{force} = W_\text{box} - F_B = 39240\,\text{N} - 39240\,\text{N} = 0\,\text{N}\). So, the force needed to fully submerge the floating cubical box is 0 N.

Key concepts

Buoyant Force

When you place an object in a fluid, like water, you may notice it seems to either float, sink, or bob somewhere in between. This behavior is explained by a force called the buoyant force. The buoyant force is an upward force exerted by the fluid that opposes the weight of an object submerged in it. According to Archimedes' Principle, the magnitude of this buoyant force is equal to the weight of the fluid displaced by the object. When a cubical box floats with half of its volume submerged, the buoyant force exactly balances the weight of the box. This means the upward push from the water is equal to the downward pull of gravity on the box. So, if you want to calculate it, you need to know:

• The volume of the submerged part of the object
• The density of the fluid
• The acceleration due to gravity

In our example, the buoyant force was calculated as 39240 N, showing how much water weight is displaced by the submerged half of the box.

Density

Density is a crucial concept when understanding why objects float or sink. Density is defined as the mass per unit volume of a substance and directly affects buoyancy.To see how density plays a role, consider water with a density of 1000 kg/m³. An object with a lower density than this tends to float, while one with a higher density sinks. In our exercise, the density of the box is half that of water, enabling it to float with half its volume submerged. You can calculate the density ratio of an object by comparing it to the density of the fluid it's in:

• If the ratio (\(f = \rho / \rho_w\)) is less than or equal to 1, the object floats.
• If the ratio is greater than 1, the object sinks.

Understanding density helps predict whether an object will sink or float based on its density relative to the fluid it is submerged in.

Submerged Volume

Submerged volume refers to the part of an object's volume that is underwater when it is placed in a fluid. It is an essential component in calculating buoyant force according to Archimedes' Principle.For an object partially submerged, like our cubical box with sides of 2 m and half submerged, we calculate submerged volume as:\[V_\text{submerged} = \text{Base Area} \times \text{Submersion Depth}\]In our scenario, since half the cube's volume is underwater, the submerged volume is half the cube's total volume, yielding 4 m³. This submerged volume displaces an equal volume of water, which is key to calculating the buoyant force.To fully submerge an object, the total volume must be submerged. This is a crucial factor when determining the forces required to submerge an object completely.

Gravity

Gravity is a force that attracts two bodies towards each other, and on Earth, it gives weight to physical objects. When discussing buoyancy, gravity is essential because it creates the weight of the object that the buoyant force needs to counteract. It is due to gravity that objects exert a force downward proportional to their mass and the gravitational acceleration:\[W_\text{object} = m \cdot g\]Where:

• \(W_\text{object}\) is the weight of the object
• \(m\) is the mass of the object
• \(g\) is the acceleration due to gravity, approximately 9.81 m/s² on Earth

In our problem, gravity ensures that the weight of the box and the buoyant force are balanced, allowing the box to float half-submerged. Additionally, when attempting to fully submerge the box, gravity's influence remains critical because it determines the weight we need to counteract with an equal or greater force.