A block of wood floats in water with 40mm projecting above the water surface. When placed in glycerin of sp gr 1.35, the block projects 70mm above the surface of that liquid. Determine the sp gr of wood.

Specific Gravity of Wood Calculation

Problem Statement

A block of wood floats in water and glycerin under the following conditions:

Projection above water surface: 40 mm
Projection above glycerin surface: 70 mm
Specific gravity of glycerin: 1.35

Determine the specific gravity of the wooden block.

Solution

1. Define Equilibrium Conditions

\[ \text{Weight of wood} = \text{Weight of displaced liquid} \] Weight of wooden block: \[ W = S \times 9810 \times A \times h \] Weight of water displaced: \[ W_W = 9810 \times A \times (h – 0.04) \] Weight of glycerin displaced: \[ W_G = 1.35 \times 9810 \times A \times (h – 0.07) \]

2. Equating Weight in Water

\[ S \times 9810 \times A \times h = 9810 \times A \times (h – 0.04) \] \[ S = \frac{h – 0.04}{h} \]

3. Equating Weight in Glycerin

\[ 9810 \times A \times (h – 0.04) = 1.35 \times 9810 \times A \times (h – 0.07) \] Solving for \( h \): \[ h = 0.155 \text{ m} = 155 \text{ mm} \]

4. Calculate Specific Gravity of Wood

\[ S = \frac{h – 0.04}{h} \] \[ S = \frac{0.155 – 0.04}{0.155} \] \[ S = 0.74 \]

Final Result:

Specific gravity of the wooden block: 0.74

Explanation

1. Floating Equilibrium:
A floating object displaces a volume of liquid equal to its own weight. Since the wooden block is floating, the weight of the wood is equal to the weight of the displaced liquid.

2. Use of Two Liquids:
The block projects differently in water and glycerin due to their different densities. Since glycerin is denser than water, the block displaces less glycerin than water to maintain equilibrium.

3. Calculation of Specific Gravity:
The height of the block is unknown, so we first set up equilibrium conditions in both water and glycerin. By solving for the total height \( h \) and using it in the specific gravity equation, we determine \( S \).

4. Verification:
The final value \( S = 0.74 \) is reasonable for a wooden block, confirming that the solution is accurate.

Physical Meaning

1. Role of Specific Gravity in Floating Objects:
The specific gravity of an object determines how much of it will be submerged in a liquid. If \( S < 1 \), the object floats; if \( S > 1 \), it sinks.

2. Application in Shipbuilding:
Engineers use this principle when designing ships to ensure they float with the desired portion above the water level.

3. Industrial Applications:
In industries, materials are selected based on their specific gravity to ensure they float or sink as needed in various processes.