A metal wire 1.0mm in diameter and 2.0cm long hangs 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 from the material?
To find the value of Young's Modulus (Y) from the given information, we can use the formula:
Y = (F/A) / (ΔL/L)
where:
- Y is Young's Modulus
- F is the force applied (weight of the mass)
- A is the cross-sectional area of the wire
- ΔL is the change in length of the wire
- L is the original length of the wire
1. Calculate the cross-sectional area (A) of the wire:
A = πr^2
= π(0.5mm)^2
= 3.14 * (0.5mm)^2
= 0.785 mm^2
2. Convert the diameter to meters:
d = 1.0mm = 0.001m
3. Calculate the original length (L) of the wire:
L = 2.0cm = 0.02m
4. Calculate the force applied (F):
F = m * g
= 6.0kg * 9.8 m/s^2
= 58.8 N
5. Calculate the change in length (ΔL) of the wire:
ΔL = 1.4mm = 0.0014m
6. Now, substitute the calculated values into the formula for Young's Modulus:
Y = (F/A) / (ΔL/L)
= (58.8 N) / (0.785 mm^2) / (0.0014m / 0.02m)
Simplifying the equation further, we get:
Y ≈ 1.18 × 10^11 N/m²
Therefore, the value of Young's Modulus for the material of the wire is approximately 1.18 × 10^11 N/m².