Final answer:
Using Hooke's Law, the force needed to stretch a similar piece of wire 1.87 cm is 48.90 N. If 21.3 N is applied to the same wire, it would stretch to 0.814 cm, assuming that measurements are in the same units and the same elasticity applies.
Step-by-step explanation:
The problem involves understanding Hooke's Law, which states that the stretch (deformation) of an elastic material is directly proportional to the force applied to it, provided the material is not stretched beyond its elastic limit. In this formula, F = k × Δx, F represents the force applied, Δx is the change in length (stretch), and k is the spring constant of the material, which remains constant for a given elasticity.
Part a
For the first part of the question, since a 17.0-N force stretches the wire 0.650 cm, we can use this to find the spring constant (k) as k = F / Δx. If another force produces a stretch of 1.87 cm in a similar wire, we can find the new force using the same k. We assume k is the same since the wire is similar.
k = 17.0 N / 0.650 cm = 26.15 N/cm
Now, to find the force that will stretch the new piece of wire 1.87 cm:
F = k × Δx' = 26.15 N/cm × 1.87 cm = 48.90 N
Part b
For the second part, if a force of 21.3 N is applied, we use k to find the new Δx.
Δx = F / k = 21.3 N / 26.15 N/cm = 0.814 cm
The student was correct that 0.814 is not the answer, but it is important to note that this solution matches the initial setting if the units are consistent. If we are expecting a different unit for length, such as meters, conversion is necessary. However, given the information here, this would be the correct calculation for stretch in centimeters.