Waves, solving Energy with position?

[sjmacewan]

Howdy,

This is only for a second year course, but i didn't think it would get answered in the intro forums...

Ok, the wording of this question doesn't make much sense to me, but I assure you that this is what it says...
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Demonstrate how

$$x=A \cos (\omega t+\phi)$$

can be the answer for

$$E=\frac{1}{2}m\frac{dx}{dt}^2 + \frac{1}{2}kx^2$$

********************************************

I just don't see how this could possibly work. Any ideas would be helpful

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As for my thoughts, I figured that either the question was written wrong, or that using the simplified energy eq'n (E=kA^2) would help...but I get nowheres using that still.

 

[Doc Al]

Demonstrate how

x=Acos(wt+phi)

can be the answer for

E=1/2m(d^2x/dt^2)^2 + 1/2k(x^2)

There's a typo in that last equation. d^2x/dt^2 is the acceleration, not the speed.

 

[sjmacewan]

thanks for pointing that out, it was supposed to be velocity so i edited it to say such. I was mixing two questions together for a second...i still don't know what to do though

 

[Doc Al]

Just plug your equation for x into the energy equation and see if it makes sense as a solution.

 

[sjmacewan]

no, that would just complicate things. when i actually subsititute the derivative in the first eq'n i can simplify it easily to $$E= \frac{1}{2} k A^2$$ but that doesn't answer the question...what it seems to want is for you to show how $$x=A \cos (\omega t+\phi)$$ actually IS the answer for $$E=\frac{1}{2}m\frac{dx}{dt}^2 + \frac{1}{2}kx^2$$

which just sounds absurd to me.

 

[sjmacewan]

hmmmm 40+ views and nothing. I personally think that the question must be wrong, but I'll continue to check back in case anyone sees a relationship that I'm missing between the two expressions

 

[Tomsk]

Solve it as a differential equation? You have $$\frac{dx}{dt}^2$$ and $$x^2$$, so if you try $$x=A'e^{\lambda t}$$ it should work. Just a suggestion, I'm not 100% on this.

 

[Doc Al]

I'm a bit puzzled why you ignored my advice to just plug it in and see if it satisfies the equation. That's the simplest way to verify a proposed solution to any equation! And that seems to be all you are asked to do: Just demonstrate that the given function of x is a solution to the energy equation. And it is!

 

[sjmacewan]

I'm sorry, I guess I just misunderstood what the question was asking really...
if you just plug it in and work though it you get the $$e=\frac{1}{2}kA^2$$ equation that one would expect, I just didn't think that THAT was what the question wanted.

Either way, sorry for ignorning your advise, but what you said is the only thing that I myself though about doing, and I wasn't convinced that was right.