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 aim to determine the ratio of the granite volume submerged in water to that in mercury, considering a granite piece floating at the water-mercury interface.
Step 1: Buoyancy Principle The solution relies on the buoyancy principle: a floating object's buoyant force equals the weight of the displaced fluid. The granite's weight is counterbalanced by water and mercury's buoyant forces. These forces are: - Water: Buoyant force equals the weight of the granite volume in water (\( V_1 \)). \[ F_{\text{buoyancy, water}} = \rho_1 g V_1 \] - Mercury: Buoyant force equals the weight of the granite volume in mercury (\( V_2 \)). \[ F_{\text{buoyancy, mercury}} = \rho_2 g V_2 \]
Step 2: Floating Condition Floating requires the granite's weight to equal the total buoyant force: \[ F_{\text{weight}} = F_{\text{buoyancy, water}} + F_{\text{buoyancy, mercury}} \] Granite's weight is \( \rho g V \), where \( V \) is the total volume. Thus: \[ \rho g V = \rho_1 g V_1 + \rho_2 g V_2 \] The total granite volume is the sum of volumes in water and mercury: \[ V = V_1 + V_2 \]
Step 3: Ratio Calculation We need the ratio \( \frac{V_1}{V_2} \). Using the buoyant force equation, substitute and simplify: \[ \rho g V = \rho_1 g V_1 + \rho_2 g V_2 \] \[ \rho V = \rho_1 V_1 + \rho_2 V_2 \] Solve for \( V_1 \) and \( V_2 \): \[ V_1 = \frac{\rho_2 - \rho}{\rho_2 - \rho_1} V \] \[ V_2 = \frac{\rho - \rho_1}{\rho_2 - \rho_1} V \] Therefore, the ratio \( \frac{V_1}{V_2} \) is: \[ \frac{V_1}{V_2} = \frac{\rho_2 - \rho}{\rho - \rho_1} \] The answer is: \[ \boxed{\frac{\rho_2 - \rho}{\rho - \rho_1}} \]