A steal wire of cross-section area `3xx10^(-6) m^(2)` can withstand a maximum strain of `10^(-3)` .Young's modulus of steel is `2xx10^(11) Nm^(-2)` .The maximum mass this wire can hold is

Options

A. 40kg
B. 60kg
C. 80kg
D. 100kg

Explanation

To find the maximum mass that the steel wire can hold, we can follow these steps:

### Step 1: Understand the relationship between stress, strain, and Young's modulus
The relationship is given by the formula:
\[ \text{Stress} = \text{Young's Modulus} \times \text{Strain} \]
Where:
- Stress (σ) = Force (F) / Area (A)
- Strain (ε) = Change in length/original length

### Step 2: Write the expression for stress in terms of force
From the definition of stress, we can express it as:
\[ \sigma = \frac{F}{A} \]
Where:
- F is the force applied on the wire (which will be equal to the weight of the mass hanging from it, i.e., F = mg)
- A is the cross-sectional area of the wire.

### Step 3: Set up the equation using the given values
We can set the stress equal to Young's modulus times the strain:
\[ \frac{mg}{A} = Y \times \epsilon \]
Where:
- \( Y \) is the Young's modulus of steel \( = 2 \times 10^{11} \, \text{N/m}^2 \)
- \( \epsilon \) is the maximum strain \( = 10^{-3} \)
- \( A \) is the cross-sectional area \( = 3 \times 10^{-6} \, \text{m}^2 \)

### Step 4: Rearrange the equation to solve for mass (m)
Rearranging the equation gives us:
\[ m = \frac{Y \times \epsilon \times A}{g} \]
Where \( g \) is the acceleration due to gravity, approximately \( 10 \, \text{m/s}^2 \).

### Step 5: Substitute the values into the equation
Substituting the known values:
\[ m = \frac{(2 \times 10^{11} \, \text{N/m}^2) \times (10^{-3}) \times (3 \times 10^{-6} \, \text{m}^2)}{10 \, \text{m/s}^2} \]

### Step 6: Calculate the mass
Calculating the numerator:
\[ (2 \times 10^{11}) \times (10^{-3}) \times (3 \times 10^{-6}) = 6 \times 10^{2} = 600 \]
Now, divide by \( g \):
\[ m = \frac{600}{10} = 60 \, \text{kg} \]

### Final Answer
The maximum mass that the wire can hold is \( 60 \, \text{kg} \).

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