Question

A piece of granite floats at the interface of mercury and water contained in a beaker as shown in the figure. If the densities of granite, water, and mercury are \( \rho \), \( \rho_1 \), and \( \rho_2 \) respectively, the ratio of the volume of granite in water to the volume of granite in mercury is:

• A. \( \frac{\rho_2 - \rho}{\rho - \rho_1} \)
• B. \( \frac{\rho_2 + \rho}{\rho_1 + \rho} \)
• C. \( \frac{\rho_2}{\rho} \)
• D. \( \frac{\rho_1}{\rho_2} \)

Hint:

In problems involving floating objects and buoyant forces, remember that the total weight of the object is balanced by the buoyant force from the displaced fluids. Use this relationship to set up equations for the volumes displaced in each fluid.

Answer 1

The Correct Option is A

We are asked to find the ratio of the volume of granite in water to the volume of granite in mercury, given that the piece of granite is floating at the interface between water and mercury.
Step 1: Understanding the Concept of Buoyancy To solve this problem, we need to use the principle of buoyancy, which states that the buoyant force acting on a floating object is equal to the weight of the displaced fluid. The weight of the granite is balanced by the buoyant force from the water and mercury. We can write the buoyant forces for both water and mercury as follows: - For water: The buoyant force is equal to the weight of the volume of granite displaced in water. Let the volume of granite in water be \( V_1 \). The buoyant force from the water is: \[ F_{\text{buoyancy, water}} = \rho_1 g V_1 \] - For mercury: The buoyant force is equal to the weight of the volume of granite displaced in mercury. Let the volume of granite in mercury be \( V_2 \). The buoyant force from mercury is: \[ F_{\text{buoyancy, mercury}} = \rho_2 g V_2 \]
Step 2: Using the Condition of Floating The condition for floating is that the weight of the granite is balanced by the total buoyant force, i.e., \[ F_{\text{weight}} = F_{\text{buoyancy, water}} + F_{\text{buoyancy, mercury}} \] The weight of the granite is \( \rho g V \), where \( V \) is the total volume of granite. Hence, the total buoyant force is: \[ \rho g V = \rho_1 g V_1 + \rho_2 g V_2 \] Since the volume of the granite is divided between water and mercury, we have: \[ V = V_1 + V_2 \]
Step 3: Expressing the Ratio Now, we need to find the ratio of the volume of granite in water to the volume of granite in mercury. This is given by: \[ \frac{V_1}{V_2} \] Using the equation of buoyant force balance, we substitute the volumes and densities: \[ \rho g V = \rho_1 g V_1 + \rho_2 g V_2 \] Simplifying: \[ \rho V = \rho_1 V_1 + \rho_2 V_2 \] Now, solve for \( V_1 \) and \( V_2 \) in terms of the total volume \( V \): \[ V_1 = \frac{\rho_2 - \rho}{\rho_2 - \rho_1} V \] \[ V_2 = \frac{\rho - \rho_1}{\rho_2 - \rho_1} V \] Thus, the ratio \( \frac{V_1}{V_2} \) is: \[ \frac{V_1}{V_2} = \frac{\rho_2 - \rho}{\rho - \rho_1} \] Thus, the correct answer is: \[ \boxed{\frac{\rho_2 - \rho}{\rho - \rho_1}} \]