A metal wire 1.0mm in diameter and 2.0cm long hands vertically with a 6.0kg mass suspended from it. If the wire stretches 1.4mm under the tension, what is the value of Young's modulus for the material?
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.