Young's Modulus (Y) 20 X 1010 N/m2
Shear Modulus (S) 8.1 X 1010 N/m2
Bulk Modulus (B) 16 X 1010 N/m2
The table to the above represents various properties of steel. You have steel wire 4.9 meters in length that stretches 0.16 cm when subjected to a force of 400 N.
What would the diameter of the wire be if you wanted the wire to stretch 0.06 cm less when subjected to this same force?
You want the amount of stretching to decrease from 0.16 cm to 0.10 cm. That is 5/8 of the orginal stetching. To make that happen, the stress nust be decreased by a factor of 5/8. Since the load is the same, the area must increase by a factor of 8/5. The diamenter must increase by a factor sqrt(8/5) = 1.265
Now we have to figure out the original wire diameter -- the one that stretched 0.16 cm.
The wire that stretched 0.16 cm had an area A given by
F/A = Y * (deltaL)/L = 20*10^10*0.16*10^-2/4.9 = 6.53*10^7 N/m^2
A = 6.13*10^-6 m^2
D = sqrt[4A/pi} = 2.79*10^-3 m = 2.79 mm
The new wire diamneter is 1.265 times 2.79 mm, or about 3.5 mm
To find the diameter of the wire, we need to use Young's modulus (Y), the original length of the wire (L), the original stretching (ΔL_1), and the desired change in stretching (ΔL_2).
The formula to calculate the change in stretching (ΔL) is:
ΔL = (F * L) / (Y * A)
where,
F is the force applied to the wire
L is the original length of the wire
Y is the Young's modulus of the steel
A is the cross-sectional area of the wire
First, let's convert the given stretching values from centimeters (cm) to meters (m):
ΔL_1 = 0.16 cm = 0.16 / 100 = 0.0016 m
ΔL_2 = 0.06 cm = 0.06 / 100 = 0.0006 m
We know the original length of the wire (L) is 4.9 m and the force applied (F) is 400 N.
Now, let's calculate the cross-sectional area (A) using the original stretching (ΔL_1):
ΔL_1 = (F * L) / (Y * A)
A = (F * L) / (Y * ΔL_1)
Substituting the given values:
A = (400 N * 4.9 m) / (20 x 10^10 N/m^2 * 0.0016 m)
Calculating A:
A ≈ 0.00012 m^2
Next, let's calculate the new diameter (d) using the desired change in stretching (ΔL_2):
ΔL_2 = (F * L) / (Y * A)
d = √[(4 * ΔL_2 * Y * A) / π]
Substituting the given values:
d = √[(4 * 0.0006 m * 20 x 10^10 N/m^2 * 0.00012 m^2) / π]
Calculating d:
d ≈ 0.0057 m
Finally, let's convert the diameter (d) from meters to centimeters:
d = 0.0057 m * 100 = 0.57 cm
Therefore, the diameter of the wire would be approximately 0.57 cm if you want the wire to stretch 0.06 cm less when subjected to the same force.