A steel wire of diameter 2 mm has a breaking strength of `4xx10^(5)`N. What is the breaking strength of similar steel wire of diameter `1.5 mm` ?

Options

A. `2.3xx10^(5)N`
B. `2.6xx10^(5)N`
C. `3xx10^(5)N`
D. `1.5xx10^(5)N`

Explanation

To find the breaking strength of a steel wire with a diameter of 1.5 mm, given that a steel wire of diameter 2 mm has a breaking strength of \(4 \times 10^5\) N, we can follow these steps:

### Step 1: Understand the relationship between breaking strength and diameter
The breaking strength (F) of a wire is directly proportional to the cross-sectional area (A) of the wire. The area of a circular cross-section is given by the formula:
\[ A = \frac{\pi D^2}{4} \]
where \(D\) is the diameter of the wire.

### Step 2: Establish the proportionality
Since breaking strength is proportional to the area, we can express this relationship as:
\[ F \propto D^2 \]
This means that if we have two wires, the ratio of their breaking strengths is equal to the ratio of the squares of their diameters:
\[ \frac{F_2}{F_1} = \left(\frac{D_2}{D_1}\right)^2 \]

### Step 3: Assign known values
Let:
- \(F_1 = 4 \times 10^5 \, \text{N}\) (breaking strength of the first wire)
- \(D_1 = 2 \, \text{mm}\) (diameter of the first wire)
- \(D_2 = 1.5 \, \text{mm}\) (diameter of the second wire)
- \(F_2\) = breaking strength of the second wire (unknown)

### Step 4: Substitute the values into the proportionality equation
Using the values we assigned:
\[ \frac{F_2}{4 \times 10^5} = \left(\frac{1.5}{2}\right)^2 \]

### Step 5: Calculate the right side of the equation
Calculate \(\left(\frac{1.5}{2}\right)^2\):
\[ \frac{1.5}{2} = 0.75 \]
\[ (0.75)^2 = 0.5625 \]

### Step 6: Solve for \(F_2\)
Now substitute back into the equation:
\[ \frac{F_2}{4 \times 10^5} = 0.5625 \]
Multiply both sides by \(4 \times 10^5\):
\[ F_2 = 0.5625 \times 4 \times 10^5 \]
Calculating this gives:
\[ F_2 = 2.25 \times 10^5 \, \text{N} \]

### Step 7: Final answer
The breaking strength of the steel wire with a diameter of 1.5 mm is:
\[ F_2 = 2.25 \times 10^5 \, \text{N} \]

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