A force F is required to break a wire of length L and radius r. Find the force required to break a wire of the same material, length 2L and radius 4r.

Answer

To solve the problem step by step, we need to understand the relationship between the breaking force, the area of cross-section, and the breaking stress of the material.

### Step 1: Understand the relationship between breaking force and area
The breaking force \( F \) required to break a wire is given by the formula:
\[ F = \text{Breaking Stress} \times \text{Area of Cross Section} \]
Breaking stress is a constant for a given material.

### Step 2: Calculate the area of cross-section for both wires
The area of cross-section \( A \) of a wire with radius \( r \) is given by:
\[ A = \pi r^2 \]
For the first wire with radius \( r \):
\[ A_1 = \pi r^2 \]
For the second wire with radius \( 4r \):
\[ A_2 = \pi (4r)^2 = \pi \times 16r^2 = 16\pi r^2 \]

### Step 3: Relate the breaking forces of the two wires
Let \( F_1 \) be the breaking force for the first wire (length \( L \) and radius \( r \)), which is given as \( F \).
Let \( F_2 \) be the breaking force for the second wire (length \( 2L \) and radius \( 4r \)).

Using the relationship between breaking force and area:
\[ \frac{F_2}{F_1} = \frac{A_2}{A_1} \]
Substituting the areas we calculated:
\[ \frac{F_2}{F} = \frac{16\pi r^2}{\pi r^2} \]

### Step 4: Simplify the equation
The \( \pi r^2 \) terms cancel out:
\[ \frac{F_2}{F} = 16 \]

### Step 5: Solve for \( F_2 \)
Now, we can express \( F_2 \) in terms of \( F \):
\[ F_2 = 16F \]

### Final Answer
The force required to break the wire of length \( 2L \) and radius \( 4r \) is:
\[ F_2 = 16F \]

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