To find the value of Young's modulus for the material, we can use the equation:
Young's modulus = (Stress / Strain)
First, let's calculate the stress.
Stress is defined as the force applied per unit area and is given by the formula:
Stress = force / area
Given:
Diameter of the wire (d) = 1.0 mm = 1.0 × 10^(-3) m
Length of the wire (L) = 2.0 cm = 2.0 × 10^(-2) m
Force (F) = mass × acceleration due to gravity (g) = 6.0 kg × 9.8 m/s^2 = 58.8 N
Area (A) = π × (radius of wire)^2
= π × (diameter of wire / 2)^2
= π × (1.0 × 10^(-3) m / 2)^2
= π × (0.5 × 10^(-3))^2
= π × 0.25 × 10^(-6) m^2
Substituting the values, we get:
A = 3.14 × 0.25 × 10^(-6) m^2
= 7.85 × 10^(-7) m^2
Now, we can calculate the stress:
Stress = F / A
= 58.8 N / 7.85 × 10^(-7) m^2
Next, let's calculate the strain.
Strain is the ratio of change in length to the original length and is given by:
Strain = change in length / original length
= (final length - initial length) / initial length
= (L - Lo) / Lo
Given:
Change in length = 1.4 mm = 1.4 × 10^(-3) m
Original length (Lo) = 2.0 cm = 2.0 × 10^(-2) m
Substituting the values:
Strain = (1.4 × 10^(-3) m) / (2.0 × 10^(-2) m)
Now, we have both the stress and strain. We can calculate Young's modulus using the formula:
Young's modulus = Stress / Strain
Substituting the values we calculated earlier, we get:
Young's modulus = (Stress) / (Strain)
= [58.8 N / (7.85 × 10^(-7) m^2)] / [(1.4 × 10^(-3) m) / (2.0 × 10^(-2) m)]
Simplifying the expression gives the value of Young's modulus for the material.