A large drop of oil whose density is less than that of water, floats up through a column of water assume that the oil an the water do not mix. The coefficient of viscosity of the oil is `eta_(@)` and that of water is `eta_(W)` the velocity of the drop will depend on

Options

A. both `eta_(@)` and `eta_(W)`
B. `eta_(@)` only
C. `eta_(W)` only
D. neither `eta_(@)` nor `eta_(W)`

Answer

To determine the factors on which the velocity of a large drop of oil floating through a column of water depends, we can analyze the situation step by step.

### Step 1: Understanding the Scenario
We have a large drop of oil that is less dense than water, and it is floating upwards through a column of water. The oil and water do not mix, which means we can treat them as two separate fluids.

**Hint:** Consider the properties of the fluids involved, especially their densities and viscosities.

### Step 2: Identifying Forces Acting on the Drop
When the oil drop moves through the water, it experiences several forces:
- **Buoyant Force:** This is the upward force exerted by the water on the oil drop, which is determined by the volume of the drop and the density of the water.
- **Gravitational Force:** This is the downward force due to the weight of the oil drop, which depends on its volume and density.
- **Viscous Drag Force:** This is the resistance force that the water exerts on the oil drop as it moves through it, which depends on the viscosity of the water.

**Hint:** Remember that the buoyant force and gravitational force will determine whether the drop rises or falls.

### Step 3: Analyzing Viscosity
The coefficient of viscosity (η) is a measure of a fluid's resistance to flow. In this case, since the oil is moving through water, the relevant viscosity is that of the water (η_W). The viscosity of the oil (η_@) does not affect the motion of the drop because the oil itself is not flowing; it is the water that is providing the resistance.

**Hint:** Focus on the fluid in which the drop is moving to determine the relevant viscosity.

### Step 4: Applying Stokes' Law
For small spherical objects moving through a viscous fluid, Stokes' Law can be applied, which states that the viscous drag force (F_d) is given by:
\[ F_d = 6 \pi r \eta_W v \]
where:
- \( r \) is the radius of the drop,
- \( \eta_W \) is the viscosity of the water,
- \( v \) is the velocity of the drop.

In equilibrium (when the drop moves at a constant velocity), the buoyant force equals the weight of the drop plus the viscous drag force.

**Hint:** Consider the balance of forces acting on the drop to understand how velocity is determined.

### Step 5: Conclusion
From the analysis, we conclude that the velocity of the oil drop floating through water is primarily dependent on the viscosity of the water (η_W) and not on the viscosity of the oil (η_@). Therefore, the velocity of the drop will depend on η_W.

**Final Answer:** The velocity of the drop will depend on η_W (the viscosity of water).