A marble of mass `x` and diameter `2r` is gently released in a tall cylinder containing honey. If the marble displaces mass `y(ltx)`of the liquid, then the terminal velocity is proportional to

Options

A. `x+y`
B. `x-y`
C. `(x+y)/r`
D. `(x-y)/r`

Explanation

To solve the problem, we need to determine the relationship between the terminal velocity of the marble and the given parameters: mass of the marble (x), mass of the liquid displaced (y), and the radius of the marble (r).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a marble of mass `x` and diameter `2r`, which means the radius `r = diameter/2`.
- The marble is released in honey, and it displaces a mass `y` of the liquid.
- Since `y < x`, the marble is denser than honey.

2. **Volume of the Marble**:
- The volume `V` of the marble can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]

3. **Using the Principle of Buoyancy**:
- According to Archimedes' principle, the weight of the liquid displaced by the marble is equal to the weight of the marble when it reaches terminal velocity.
- The weight of the marble is `x * g` (where `g` is the acceleration due to gravity).
- The weight of the displaced liquid is `y * g`.

4. **Setting Up the Equation**:
- At terminal velocity, the forces acting on the marble balance out, so we can write:
\[ x \cdot g = y \cdot g + F_d \]
- Here, `F_d` is the drag force acting on the marble.

5. **Drag Force and Terminal Velocity**:
- The drag force can be expressed using Stokes' law:
\[ F_d = 6 \pi \eta r v_t \]
- Where `v_t` is the terminal velocity and `η` is the viscosity of the honey.

6. **Equating Forces**:
- Rearranging the equation gives:
\[ F_d = (x - y) \cdot g \]
- Substituting for `F_d` gives:
\[ 6 \pi \eta r v_t = (x - y) \cdot g \]

7. **Solving for Terminal Velocity**:
- Rearranging for `v_t` yields:
\[ v_t = \frac{(x - y) \cdot g}{6 \pi \eta r} \]

8. **Identifying Proportionality**:
- From the equation, we can see that terminal velocity `v_t` is proportional to:
\[ v_t \propto \frac{x - y}{r} \]

### Conclusion:
The terminal velocity of the marble is proportional to the difference in mass between the marble and the displaced liquid divided by the radius of the marble.