Given:
- The flood lamp stretches the wire by 0.18 mm
- The stress is proportional to the strain
To find:
- How much would it have stretched if the wire had the same length but twice the diameter

Let's begin by understanding the given information.

Stress is defined as the force per unit area and is denoted by the symbol σ (sigma). Mathematically, stress is given by:

σ = F / A

where F is the force applied and A is the area over which the force is applied.

Strain is defined as the change in length per unit length and is denoted by the symbol ε (epsilon). Mathematically, strain is given by:

ε = ΔL / L

where ΔL is the change in length and L is the original length.

From the given information, we know that the stress is proportional to the strain. This can be expressed mathematically as:

σ ∝ ε

or

σ = kε

where k is a constant of proportionality.

Now, let's apply this information to the problem at hand.

When the flood lamp is hung from the wire, it exerts a force on the wire which causes it to stretch. Let's assume that the original diameter of the wire is d and the original length is L.

From the given information, we know that the stress is proportional to the strain. Therefore, we can write:

σ = kε

where σ is the stress, k is a constant of proportionality, and ε is the strain.

The stress can be calculated using the formula:

σ = F / A

where F is the force applied and A is the cross-sectional area of the wire.

The force applied is the weight of the flood lamp, which can be calculated using the formula:

F = mg

where m is the mass of the flood lamp and g is the acceleration due to gravity.

The cross-sectional area of the wire can be calculated using the formula:

A = πd^2 / 4

where d is the diameter of the wire.

Therefore, we can write:

σ = (mg) / (πd^2 / 4)

The strain can be calculated using the formula:

ε = ΔL / L

where ΔL is the change in length and L is the original length.

From the given information, we know that the flood lamp stretches the wire by 0.18 mm. Therefore, we can write:

ε = 0.18 / L

Now, let's combine the equations for stress and strain:

σ = kε

σ = (mg) / (πd^2 / 4)

ε = 0.18 / L

Substituting the values of σ and ε, we get:

(mg) / (πd^2 / 4) = k (0.18 / L)

Simplifying, we get:

k = (mgL) / (0.18πd^2)

Now, let's use this value of k to calculate the change in length when the diameter of the wire is doubled.

When the diameter of the wire is doubled, the cross-sectional area of the wire becomes 4 times the original area. Therefore, the new diameter is 2d and the new cross-sectional area is:

A' = π(2d)^2 / 4 = 4πd^2

Using the same formula for stress,