A 4.0kg object is supported by an aluminum wire of length 2.5 m and diameter 1.8 mm
How much will the wire stretch?
I need the breakdown of how this is being solved so I can figure it out in the future. Even just a formula will do
To calculate how much the wire will stretch, you can use Hooke's Law, which states that the amount of stretch or compression of an elastic material is directly proportional to the force applied to it, multiplied by its spring constant.
The formula for Hooke's Law is as follows:
ΔL = (F * L) / (A * Y)
Where:
ΔL is the change in length of the wire
F is the force applied to the wire
L is the original length of the wire
A is the cross-sectional area of the wire
Y is the Young's modulus of aluminum
To find the change in length, you need to calculate the force applied to the wire. In this case, the force is the weight of the object, which can be calculated using the formula:
F = m * g
Where:
F is the force
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s²)
Now let's break down the calculations for your specific example:
1. Calculate the force:
F = 4.0 kg * 9.8 m/s²
F = 39.2 N
2. Calculate the cross-sectional area:
The cross-sectional area of a wire can be calculated using the formula:
A = π * (d/2)²
where:
A is the cross-sectional area
d is the diameter of the wire
Plugging in the values:
A = π * (1.8 mm / 2)²
A = π * (0.0009 m)²
A ≈ 0.00000254 m²
3. Determine the Young's modulus of aluminum:
The Young's modulus of a material is a measure of its stiffness. For aluminum, it is approximately 70 GPa (gigapascals) or 70 x 10^9 Pa.
4. Calculate the change in length:
ΔL = (39.2 N * 2.5 m) / (0.00000254 m² * 70 x 10^9 Pa)
ΔL ≈ 0.580 x 10^-6 m
So, the wire will stretch by approximately 0.580 micrometers (×10^-6 m) when a 4.0 kg object is supported by it.