Final answer:
Compressing a spring to a quarter of its relaxed length requires four times the energy used to compress it to half its length. Since it took 24 J to compress the spring to half, it will take an additional 72 J (for a total of 96 J) to compress it to a quarter of its relaxed length.
Step-by-step explanation:
To figure out how much more energy is required to compress the spring to a quarter of its relaxed length, compared to half its length, we can use Hooke's Law, which states that the potential energy stored in a compressed spring is proportional to the square of its compression distance. This means that compressing the spring to half its length requires energy proportional to that length squared. If it took 24 Joules to compress the spring to half its original length, compressing it to a quarter of its original length would involve squaring half of the already halved length, which would be equivalent to four times the initial length squared.
Since energy is proportional to the square of the compression, to compress the spring to a quarter its length requires four times the energy than that needed to compress it to half its length. Therefore, if 24 Joules compressed it to half, then 96 Joules in total are needed to compress the spring to a quarter its relaxed length. You've already spent 24 Joules, so the additional energy required is 96 Joules - 24 Joules, which is 72 Joules more.