A circular steel wire 2.00 m is to stretch no more than 0.25 cm when a tensile force of 400 N is applied to each end of wire. What minimum diameter is required for the wire?
To find the minimum diameter required for the wire, we can use Hooke's Law, which states that the elongation or extension of a spring or wire is directly proportional to the applied force and inversely proportional to the cross-sectional area.
First, let's convert the given elongation from centimeters to meters:
0.25 cm = 0.25/100 = 0.0025 m
Now let's consider the formula to find the elongation of a wire or spring:
ΔL = (F * L) / (π * r^2 * E)
Where:
ΔL is the elongation (0.0025 m)
F is the force (400 N)
L is the original length of the wire (2.00 m)
r is the radius of the wire (we want to find this)
E is the Young's modulus of the material (a property of the material, given separately)
Since we are dealing with a circular wire, the radius (r) is half the diameter (d). So, let's rewrite the formula in terms of the diameter:
ΔL = (F * L) / (π * (d/2)^2 * E)
Rearranging the formula to solve for the diameter (d), we get:
d = √((4 * F * L) / (π * ΔL * E))
Now we can substitute the given values into the formula:
d = √((4 * 400 * 2.00) / (π * 0.0025 * E))
To find the minimum diameter, we need to know the Young's modulus (E) of the material, which is not provided in the question. The Young's modulus differs for different materials.
Once you have the value of the Young's modulus for the specific material being used, you can substitute it into the equation to find the minimum diameter required for the wire.
area = pi d^2/4
length = 2 m
change in length = .0025 m
change in length/length = .0025/2
= tensile stress/E = 400/(pi d^2/4 E)
so
.00125 = 400/ (pi d^2/4 E)