Direct/Inverse Squared relationships problems???

[Descarteo]

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Can anyone pls solve (and explain) this direct relationship problem:

"Elastic potential energy in a spring is directly proportional to the square of the displacement of one end of the spring from its rest position while the other end remains fixed. If the elastic potential energy in the spring is 100J when it is compressed to half its rest length, what is its energy when it is compressed to one fourth its rest length?"

a.50J
b.150J
c.200J
d.225J


I underlined important terms cos i have trouble manipulating squared and inverse squared relationships, i.e. if one increases by a factor of 4, the other decreases or increases by a factor of sqrt 4, and all that stuff.

Explain your answers please.

Thanks in advance.

 

[physics junkie]

First, consider posting in the proper forum next time. There is a Q&A forum.

Elastic Potential Energy is given by the equation: E = (1/2)*k*x^2

k = spring constant
x= displacement from the equilibrium length of the spring(the length of the spring from end to end when it is just sitting there on a table)

For simplicity of calculation let's just assume it is a 1 meter long spring and that a displacement of 0.5m represents its length being shortened by half and that a displacement of 0.75m means its being cut to 1/4th its original length.

Using the equation above...

100 = (0.5)*k*(0.5)^2
= 0.5*k*0.25*x^2
= 0.125*k*x^2

Now you want to know how much energy for 0.75x

? = (0.5)*k*(0.75*x)^2
? = 0.28125*k*x^2

If you divide the two by eachother you'll notice that 0.28125/0.125 = 2.25. So you cut the spring length in half and the energy increased by 2.25 fold.

Remember, the 0.5 and 0.75 are the displacement from equilibrium spring length. As displacement increases so does spring energy.

**Edited due to error.

 

[likekekaika]

Eliminate answers first, more compressed = more PE, therefore A is wrong. The answer should be fairly simple to figure out by reverse math. Take the square root of the elastic energy when half compressed which is 10. Since the spring can fully compress to a value of 20 (2x half compressed), when compressed to one fourth, it is compressed to a value of 15. 15 * 15 is 225J. Hope this helps. All physics problems can be easily solved if drawn out on paper and all values removed except the ones needed. Then after that it is just plug and chug. What physics junkie wrote seems a bit overworked for this question because all the information you need is already in the passage but would still lead you to the right answer.

 

[physics junkie]

By the way, I wouldn't ever solve the question the way I just explained. Once you understand it you can develop some bizarre way to do the problem like likekekaika or myself. 😀