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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.
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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