Simple Harmonic Motion - Potential and Kinetic Energy?

[meganw]

Homework Statement

A 2.00 kg mass vibrates according to the equation x = 0.470 cos 8.36t, where x is in meters, and t is in seconds. Assume that x = 0.29 m.

(a) Determine the amplitude.
.470
(b) Determine the frequency.
1.331
(c) Determine the total energy.

(d) Determine the kinetic energy.

e) Determine the potential energy.

I did the first two parts (a-b), but I'm stuck on c,d & e, the questions about the Energy of the spring.

Homework Equations

PE = .5kx^2

and

KE = .5KA^2

F/x=k

The Attempt at a Solution

I've tried these 3 multiple times and I keep getting them wrong. One thing I'm not sure of, is is "x" the amplitude? I have been plugging in .29 for the "x" and .47 for the "A", but just curious if this is the reason I'm getting these wrong.

I would think, that according to these formulas, KE+PE=TOTAL ENERGY, and I thought I was doing them right. I'm at my wits end.

 

[Doc Al]

Homework Equations

PE = .5kx^2

Good.

and

KE = .5KA^2

Not good. (This implies that the KE doesn't change!)

I've tried these 3 multiple times and I keep getting them wrong. One thing I'm not sure of, is is "x" the amplitude?

No. "x" is displacement from equilibrium.

I have been plugging in .29 for the "x" and .47 for the "A", but just curious if this is the reason I'm getting these wrong.

These are correct.

I would think, that according to these formulas, KE+PE=TOTAL ENERGY,

Absolutely. What's the total energy?

Hint: Since total energy doesn't change, pick the easiest position to calculate PE + KE.

 

[Snazzy]

K.E + P.E = total energy
$$\frac{1}{2}mv^2+\frac{1}{2}kx^2=\frac{1}{2}kA^2$$
They usually give you that equation on formula sheets.

 

[meganw]

Does Ke=.5mv^2?

I don't know V, and I can't use w=v/r because I don't know R.

ALSO:

I tried to find Total Energy when PE is maxed out and v=0, but I got it wrong. I did

Total Energy=.5((9.8*2)/(.29))*(.29)^2
thats .5 * k * x^2

And Snazzy-I tried .5KA^2 as my total energy and got 2.986 and that's wrong...

 

[Snazzy]

The total energy is only given by $$\frac{1}{2}kx^2$$ only if $$x=A$$

You find A by looking at the equation of the motion because SHM follows the equation:

$$x(t)=Acos(\omega t+\phi)$$

Then you can find the potential energy at x=0.29m and then the kinetic energy at that point since you now have the total energy and the potential energy.

 

[meganw]

But I can't find KE because I don't know V and I can't find v using W because I don't have R...

I TRIED TO FIND PE AND GOT IT WRONG:

F=kx

x=.29
F=9.8*2
k=67.5862

SO

PE=.5(67.5862)(.29^2)
=2.842

Thank you so much for your help on this!
wrong

 

[Snazzy]

You don't need V or R or whatever to find the kinetic energy. You can use the fact that KE + PE = total energy

 

[meganw]

Well, x is not equal to A. So I guess I can't use the fact that .5kA^2 is total, and I keep getting PE wrong. :(

 

[Snazzy]

You get PE wrong because your value of k is wrong. F = kx at equilibrium, but x=0.29 is not the distance the spring has stretched at its equilibrium state. They usually give you the equation:

$$\omega =\sqrt{\frac{k}{m}}$$

And you CAN use the fact that $$\frac{1}{2}kA^2$$ is the total energy because A is GIVEN TO YOU IN THE EQUATION OF THE FUNCTION. The reason you got it wrong is because your value of k is wrong.

 

[meganw]

Okay, I tried it using x=.470 for my equation of the value of k, and I get k=41.7

but then I still get the wrong answer for PE...=1.753

 

[Snazzy]

I don't know why you're using the amplitude to find the value of k, use $$\omega$$ and the mass to find the value of k as shown in the equation above. The spring does not stretch to its amplitude at equilibrium, nor does it stretch to the value of x the question gives you.

 

[meganw]

I GOT THEM RIGHT! =)

Thanks, you were SO helpful! I didn't realize that you can't always use the equation of F=-kx to find k!

I'm really thankful and grateful that you stayed online and helped me through each step. Thanks again, you're a really good person. =)

 

[kdv]

I GOT THEM RIGHT! =)

Thanks, you were SO helpful! I didn't realize that you can't always use the equation of F=-kx to find k!

I'm really thankful and grateful that you stayed online and helped me through each step. Thanks again, you're a really good person. =)

You can always use F=-k x at two conditions: that the F you use is the force exerted by the spring and that the x you use is the displacement of the mass from the equilibrium position. I am not sure why you concluded that you can't always use that equation but I wanted to point that out.