Copper of fixed volume 'V' is drawn into wire of length 'l'. When this wire is subjected to a constant force 'F', the extension produced in the wire is 'Δl'. Which of the following graphs is a straight line?

Question

A 2 $\text{kg}$ mass is attached to a spring with spring constant $ k = 200, \text{N/m} $. If the mass is displaced by $ 0.1, \text{m} $, what is the potential energy stored in the spring?

Answer 1

To determine which graph represents a straight line for the given parameters, we need to analyze the relationship between the given variables based on the concepts of physics related to elasticity and material properties.

The condition given involves a copper wire of fixed volume 'V' drawn into a wire of length 'l'. When this wire is subjected to a constant force 'F', the extension produced in the wire is denoted as 'Δl'.

Hooke's Law, which describes the elasticity of materials, states that the extension Δl of a wire under tension is proportional to the original length of the wire and inversely proportional to the cross-sectional area, i.e.,

\[\Delta l = \frac{F \cdot l}{A \cdot Y}\]

where:

F is the force applied,
A is the cross-sectional area of the wire,
Y is the Young's modulus of the material (copper in this case).

Since the volume V of the wire is constant, we know the relation:

\[A \cdot l = V\]

Substituting for A in the extension formula, we have:

\[\Delta l = \frac{F \cdot l^2}{Y \cdot V}\]

From this equation, it is evident that Δl is directly proportional to l^2, meaning the relationship between Δl and l^2 is linear.

Therefore, the graph of Δl versus l^2 will be a straight line. The correct answer is:

Δl versus l^2

The analysis shows why other options are not correct:

Δl versus $\frac{1}{l^2}$: This relationship would be inverse, not linear.
Δl versus l: This graph would represent a parabolic relation when squared term is involved.
Δl versus $\frac{1}{l}$: Again, this represents an inverse non-linear relationship.

Answer 2