A block of wood weighing `71.2N` and of specific gravity `0.75` is tied by a string to the bottom of a tank of water in order to have the block totally immersed, What is the tension in the string?

Explanation

To solve the problem of finding the tension in the string holding the block of wood submerged in water, we can follow these steps:

### Step 1: Understand the Forces Acting on the Block
The block of wood experiences three forces:
- The weight of the block (W) acting downwards.
- The buoyant force (B) acting upwards.
- The tension (T) in the string acting upwards.

### Step 2: Calculate the Weight of the Block
The weight of the block is given as:
\[ W = 71.2 \, \text{N} \]

### Step 3: Determine the Specific Gravity and Density
The specific gravity (SG) of the block is given as 0.75. Specific gravity is defined as the ratio of the density of the substance to the density of water (which is approximately \(1000 \, \text{kg/m}^3\)):
\[ \text{SG} = \frac{\text{Density of block}}{\text{Density of water}} \]
Thus, the density of the block can be calculated as:
\[ \text{Density of block} = \text{SG} \times \text{Density of water} = 0.75 \times 1000 \, \text{kg/m}^3 = 750 \, \text{kg/m}^3 \]

### Step 4: Calculate the Volume of the Block
Using the weight of the block and the density, we can find the volume (V) of the block:
\[ W = \text{Density of block} \times V \times g \]
Rearranging gives:
\[ V = \frac{W}{\text{Density of block} \times g} \]
Using \( g \approx 9.81 \, \text{m/s}^2 \):
\[ V = \frac{71.2 \, \text{N}}{750 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2} \]

### Step 5: Calculate the Buoyant Force
The buoyant force (B) acting on the block can be calculated using Archimedes' principle:
\[ B = \text{Density of water} \times V \times g \]
Substituting the volume calculated in Step 4:
\[ B = 1000 \, \text{kg/m}^3 \times V \times 9.81 \, \text{m/s}^2 \]

### Step 6: Set Up the Equation of Forces
In equilibrium, the sum of the forces acting on the block is zero:
\[ B = W + T \]
Rearranging gives:
\[ T = B - W \]

### Step 7: Substitute the Values
From the previous steps, we can express the buoyant force in terms of the weight:
Using the specific gravity:
\[ B = \frac{4}{3} W \]
Now substituting this into the tension equation:
\[ T = \frac{4}{3} W - W = \frac{4W}{3} - \frac{3W}{3} = \frac{W}{3} \]

### Step 8: Calculate the Tension
Now substituting the weight:
\[ T = \frac{71.2 \, \text{N}}{3} \approx 23.73 \, \text{N} \]

Thus, the tension in the string is approximately:
\[ T \approx 23.7 \, \text{N} \]

### Final Answer
The tension in the string is approximately **23.7 N**.

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