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A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 400 N is applied to each end of the wire. What minimum diameter is required for the wire?
  • a)
    1.4 mm
  • b)
    10 mm
  • c)
    1.5 mm
  • d)
    12.4 mm
Correct answer is option 'A'. Can you explain this answer?
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To find the minimum diameter required for the wire, we can use Hooke's Law and the formula for the elongation of a wire.

1. Determining the elongation of the wire:
According to Hooke's Law, the force applied on a wire is directly proportional to its elongation. The formula for the elongation of a wire is given by:

ΔL = (F * L) / (π * r^2 * E)

where ΔL is the elongation, F is the force applied, L is the original length of the wire, r is the radius of the wire, and E is the Young's modulus of the material (steel in this case).

2. Substituting the given values:
Given:
Force applied, F = 400 N
Original length, L = 2.00 m
Elongation, ΔL = 0.25 cm = 0.0025 m (since 1 cm = 0.01 m)

We need to find the minimum diameter, which is equal to twice the radius of the wire.

3. Rearranging the formula:
We can rearrange the formula for the elongation to solve for the radius, r:

r = sqrt((F * L) / (π * ΔL * E))

4. Substituting the values and calculating:
Substituting the given values into the formula:

r = sqrt((400 * 2.00) / (π * 0.0025 * E))

The value of Young's modulus for steel is approximately 200 GPa (GigaPascals) or 200 * 10^9 Pa.

r = sqrt((400 * 2.00) / (π * 0.0025 * 200 * 10^9))

Simplifying the equation:

r = sqrt((800) / (π * 0.0025 * 200 * 10^9))

r = sqrt(800) / sqrt(π * 0.0025 * 200 * 10^9)

r ≈ 0.0286 m

5. Calculating the minimum diameter:
The minimum diameter is equal to twice the radius:

d = 2 * r

d ≈ 2 * 0.0286

d ≈ 0.0572 m

Converting the diameter to millimeters (mm):

d ≈ 57.2 mm

Therefore, the minimum diameter required for the wire is approximately 57.2 mm, which is option A.
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A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 400 N is applied to each end of the wire. What minimum diameter is required for the wire?a)1.4 mmb)10 mmc)1.5 mmd)12.4 mmCorrect answer is option 'A'. Can you explain this answer?
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