The change in length of a steel wire under an applied load can be halved by keeping all other conditions constant but using:which of the following is correct
A. a different material having a Young's modulus half that of steel
B half the cross sectional area of the steel wire
C half the length of steel wire
D half the radius of the steel wire
B. Half the cross sectional area of the steel wire
The correct answer is D. Half the radius of the steel wire.
According to Hooke's Law, the change in length of a wire is directly proportional to the applied load and inversely proportional to the cross-sectional area of the wire.
Therefore, by halving the radius of the steel wire while keeping all other conditions constant, the cross-sectional area of the wire will also be halved. This will result in a halving of the change in length for the same applied load, as per Hooke's Law.
To determine the correct answer, we need to understand the relationship between the change in length of a steel wire and the various factors. The change in length of a wire under an applied load is given by the formula:
ΔL = (F * L) / (A * E)
where:
ΔL = change in length
F = applied load
L = original length
A = cross-sectional area
E = Young's modulus
Now, let's analyze each option:
A. Using a different material having a Young's modulus half that of steel:
In this scenario, if we use a material with a Young's modulus half that of steel, the value of E would decrease. According to the formula, if the value of E decreases, the change in length (ΔL) would increase for the same applied load, leading to a larger change in length, not a halving.
B. Halving the cross-sectional area of the steel wire:
If we halve the cross-sectional area (A) of the steel wire, according to the formula, the change in length (ΔL) would increase for the same applied load. This means that halving the cross-sectional area would not cause the change in length to be halved.
C. Halving the length of steel wire:
If we halve the length (L) of the steel wire, according to the formula, the change in length (ΔL) would decrease. This means that halving the length would cause the change in length to be halved. Therefore, option C is a possible correct answer.
D. Halving the radius of the steel wire:
Since the formula involves the cross-sectional area (A = πr^2), halving the radius would reduce the cross-sectional area by a factor of 1/4. According to the formula, if the cross-sectional area decreases, the change in length (ΔL) would increase for the same applied load. Therefore, halving the radius would not cause the change in length to be halved.
Based on this analysis, option C, halving the length of the steel wire, is the correct answer.