Understanding the Forces on a Submerged Object

This problem involves a spherical object submerged in water and held down by a chain. To find the weight of the object in air, we need to understand the forces acting on it while it's submerged. When an object is submerged in a fluid, it experiences three main vertical forces:

• The object's weight (\(W_{\text{air}}\)), acting downwards. This is the force we want to find.
• The buoyant force (\(F_B\)), acting upwards. This force is exerted by the fluid (water) and is equal to the weight of the fluid displaced by the object.
• The tension in the chain (\(T\)), acting downwards, as the chain is holding the object down against its buoyancy.

Since the object is completely submerged and held stationary by the chain, it is in equilibrium. This means the sum of the upward forces equals the sum of the downward forces.

Upward force = Downward forces

\(F_B = W_{\text{air}} + T\)

We are given the tension \(T = 5.2\) kN and need to find \(W_{\text{air}}\). We can rearrange the equation to solve for \(W_{\text{air}}\):

\(W_{\text{air}} = F_B - T\)

To find \(W_{\text{air}}\), we first need to calculate the buoyant force \(F_B\).

Calculating the Buoyant Force

According to Archimedes' principle, the buoyant force on a submerged object is equal to the weight of the volume of fluid it displaces. The formula for buoyant force is:

\(F_B = \rho_{\text{fluid}} \times V_{\text{submerged}} \times g\)

Where:

• \(\rho_{\text{fluid}}\) is the density of the fluid (water). We will use the standard density of water, which is approximately \(1000 \, \text{kg/m}^3\).
• \(V_{\text{submerged}}\) is the volume of the object submerged in the fluid. Since the object is completely immersed, this is the total volume of the spherical object.
• \(g\) is the acceleration due to gravity. We will use the standard value, approximately \(9.81 \, \text{m/s}^2\).

First, let's calculate the volume of the spherical object. The diameter is \(D = 1.45\) m. The radius is \(r = D/2 = 1.45/2 = 0.725\) m.

The volume of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\).

\[V = \frac{4}{3}\pi (0.725 \, \text{m})^3\]

\[V = \frac{4}{3}\pi (0.381078125 \, \text{m}^3)\]

\[V \approx 1.6006 \, \text{m}^3\]

Now, we can calculate the buoyant force \(F_B\):

\[F_B = (1000 \, \text{kg/m}^3) \times (1.6006 \, \text{m}^3) \times (9.81 \, \text{m/s}^2)\]

\[F_B \approx 15705.89 \, \text{N}\]

To match the units of tension (kN), let's convert the buoyant force to kilonewtons:

\[F_B \approx \frac{15705.89 \, \text{N}}{1000} = 15.706 \, \text{kN}\]

Calculating the Weight in Air

Now that we have the buoyant force \(F_B\) and the chain tension \(T\), we can calculate the weight of the object in air \(W_{\text{air}}\) using the force balance equation:

\[W_{\text{air}} = F_B - T\]

We are given \(T = 5.2 \, \text{kN}\) and we calculated \(F_B \approx 15.706 \, \text{kN}\).

\[W_{\text{air}} \approx 15.706 \, \text{kN} - 5.2 \, \text{kN}\]

\[W_{\text{air}} \approx 10.506 \, \text{kN}\]

The question asks for the weight of the object 'nearly'. Our calculated value of approximately 10.506 kN is very close to 10.5 kN.

Summary of Calculation Steps

1. Identify the forces acting on the submerged object (Weight, Buoyant force, Tension).
2. Apply the condition for equilibrium (Upward forces = Downward forces): \(F_B = W_{\text{air}} + T\).
3. Rearrange to find the weight in air: \(W_{\text{air}} = F_B - T\).
4. Calculate the volume of the spherical object using its diameter.
5. Calculate the buoyant force using Archimedes' principle: \(F_B = \rho_{\text{water}} \times V \times g\).
6. Substitute the calculated buoyant force and the given tension into the equation for \(W_{\text{air}}\).
7. Determine the closest option.

Our calculated weight in air is approximately 10.506 kN, which is nearly 10.5 kN. Comparing this value to the given options, 10.5 kN is the closest value.

Revision Table: Submerged Object Forces

Additional Information on Buoyancy and Fluid Mechanics

This problem demonstrates key principles in fluid mechanics, particularly buoyancy. Buoyancy is a fundamental concept that explains why objects float or sink.

• Archimedes' Principle: States that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The direction of the force is upwards.
• Density: Density (\(\rho\)) is a measure of mass per unit volume (\(\rho = m/V\)). The density of the object relative to the fluid determines whether it will float or sink if unrestrained. If the object's density is less than the fluid's density, it will float. If it's greater, it will sink.
• Weight vs. Apparent Weight: The weight of an object in air is its true weight (\(W_{\text{air}}\)). When an object is submerged, it appears lighter because of the upward buoyant force. This apparent weight (\(W_{\text{apparent}}\)) is \(W_{\text{apparent}} = W_{\text{air}} - F_B\). In our problem, the object is denser than water (as it requires a chain to hold it down), so its weight is greater than the buoyant force, and the chain tension provides the extra downward force needed for equilibrium.

Understanding these forces and principles is crucial for solving problems involving objects in fluids, whether they are floating, sinking, or held in place by external forces like tension or compression.