An 8 kg mass is supported by an aluminum wire of length 80 cm abd diameter 2.0mm. How much will the wire stretch?
To determine how much the wire will stretch, we can use Hooke's Law, which states that the amount a spring or wire stretches is directly proportional to the force applied to it.
Hooke's Law can be expressed as:
F = k * ΔL
Where:
F is the force applied to the wire
k is the spring constant (a measure of the stiffness of the wire)
ΔL is the change in length of the wire
To find the change in length of the wire, we need to know the spring constant. The spring constant can be determined using the formula:
k = (π * (d^2) * E) / (4 * L)
Where:
d is the diameter of the wire
E is the Young's modulus of the material (for aluminum, it is approximately 70 GPa)
L is the length of the wire
Given:
Mass of the object (m) = 8 kg
Length of the wire (L) = 80 cm = 0.8 m
Diameter of the wire (d) = 2.0 mm = 0.002 m
Young's modulus for aluminum (E) = 70 GPa = 70 * 10^9 Pa
Now we can substitute the values into the equation to find the value of k:
k = (π * (0.002^2) * 70 * 10^9) / (4 * 0.8)
Simplifying this equation will give us the spring constant, k.
Once we have the spring constant, we can determine the change in length (ΔL) using Hooke's Law:
ΔL = F / k
To find the force (F) applied to the wire, we can use the equation:
F = m * g
Where:
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the values into the equation will give us the force, F.
Finally, we substitute the values of F and k into the equation for ΔL to calculate the stretch in the wire.