Greetings, If someone can explain to me using proportions (not actual physics)to

Question

Greetings,

If someone can explain to me using proportions (not actual physics)to solve this problem, by hand and without the aid of a calculator,I will rate as a lifesaver because you will literally be one! I ampracticing a MCAT physics style question and it stated that theproblem would be quicker and easier to do with proportions.

The question states: Elastic Potential energy in a spring isdirectly proportional to the square of the displacement of one endof the spring from its rest position while the other end remainsfixed. If the elastic potential energy in the spring is 100 J whenit is compressed to half its rest length, what is its energy whenit is compressed to one fourth its rest length?

Ans= 225 J.

The solution states:If we imagine a spring100 cm long at rest, thenthe initial displacement is 50 cm and the final displacement is 75cm. The displacement is increased by a factor of 1.5 thus theenergy is increased by a factor of 1.5^2. 1.5^2 is greater than1.4^2 or greater than 2. Thus, the energy is greater than 2 X100.


May someone simplify this and show how it can be done step by step?I am so confused to how the concept of proportions apply here andhave no clue what the answer solution is saying.

Thank you in advance and again will rate lifesaver for the simplestexplanation.

Explanation / Answer

The potential energy of a compressed spring is E = 1/2 kx2 E1 = 1/2 k (L/2)2   since the spring is compressed by 1/2 its length E2 = 1/2 k (3 L / 4)2   the spring is compressed by 3/4 of its length E2 / E1 = (3/4)2 /(1/2)2 = 9/4   so E2 = 9 * 100 / 4= 225 What they are attempting to say is since the potential energyis proportional to x2 the ratio of the energies is (3/4)2 to(1/2)2 since these are the relative distances by which the spring is compressed. the ratio of the energies is (3/4)2 to(1/2)2 since these are the relative distances by which the spring is compressed.

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