A wooden piece can float both in mercury (of density 13.6 gm/cc) and in water (of density 1gm/cc). The ratio of mass of mercury displaced to the mass of water displaced is

Options

A. 1
B. 13.6
C. `1/13.6`
D. `12.6/13.6`

Explanation

To solve the problem of finding the ratio of the mass of mercury displaced to the mass of water displaced by a floating wooden piece, we can follow these steps:

### Step 1: Understand the Floating Condition
When an object floats, the weight of the object is equal to the weight of the fluid displaced. This is known as Archimedes' principle.

### Step 2: Define the Variables
Let:
- \( V \) = Volume of the wooden piece (which is the same in both cases)
- \( \rho_{Hg} \) = Density of mercury = 13.6 g/cm³
- \( \rho_{w} \) = Density of water = 1 g/cm³

### Step 3: Calculate the Weight of the Wooden Piece
The weight of the wooden piece can be expressed as:
\[ W = \text{mass} \times g \]
However, since we are looking for the ratio, we can ignore \( g \) (acceleration due to gravity) as it will cancel out later.

### Step 4: Calculate the Mass of Fluid Displaced
1. **For Mercury:**
The mass of mercury displaced when the wooden piece is floating in mercury can be calculated as:
\[ \text{Mass of mercury displaced} = \rho_{Hg} \times V_{Hg} \]
where \( V_{Hg} \) is the volume of mercury displaced, which is equal to the volume of the wooden piece \( V \) when it is floating.
\[ \text{Mass of mercury displaced} = \rho_{Hg} \times V \]

2. **For Water:**
Similarly, the mass of water displaced when the wooden piece is floating in water is:
\[ \text{Mass of water displaced} = \rho_{w} \times V_{w} \]
where \( V_{w} \) is the volume of water displaced, which is also equal to the volume of the wooden piece \( V \).
\[ \text{Mass of water displaced} = \rho_{w} \times V \]

### Step 5: Set Up the Ratio
Now, we can set up the ratio of the mass of mercury displaced to the mass of water displaced:
\[ \text{Ratio} = \frac{\text{Mass of mercury displaced}}{\text{Mass of water displaced}} = \frac{\rho_{Hg} \times V}{\rho_{w} \times V} \]
The volume \( V \) cancels out:
\[ \text{Ratio} = \frac{\rho_{Hg}}{\rho_{w}} \]

### Step 6: Substitute the Densities
Now, substituting the values of the densities:
\[ \text{Ratio} = \frac{13.6 \, \text{g/cm}^3}{1 \, \text{g/cm}^3} = 13.6 \]

### Conclusion
Thus, the ratio of the mass of mercury displaced to the mass of water displaced is **13.6**.

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