Final answer:
Correct option: (d) L/3 has force constant k/3
The correct answer is that the smallest piece of the spring, with a length of L/3, will have a force constant of 3k. This is deduced from the inverse proportionality between force constant and length of a spring when it is cut.
Step-by-step explanation:
The question discusses how the force constant (k) of a spring changes when the spring is cut into pieces.
The original spring, with force constant k and length L, is divided into three pieces in the ratio 1:2:3.
To determine the force constant of each piece, we must remember that the force constant of a spring is inversely proportional to its length.
This means if you cut a spring into pieces, each piece will have a force constant that is inversely proportional to its new length.
Thus, for each piece the force constant can be calculated by dividing k by the fraction that represents the piece's length with respect to the original spring length L.
For option (a), the piece length is 2L/3, which is not one of the given ratios, making this option incorrect. Option (b) implies the smallest piece is L/6 of the total length, leading to a force constant of 6k, which is incorrect. Option (c) states the smallest piece is L/3 of the total length, suggesting the force constant to be k/3, but it would actually be 3k, which makes this option incorrect too.
Finally, option (d) also suggests that for a piece L/3 long, the force constant is k/3, which is incorrect, as the correct force constant would be 3k.
Therefore, the correct answer is: the smallest piece which is L/3 of the total length will have a force constant that is three times the original force constant, so the correct force constant is 3k.