- #1
User_Unknown9
- 1
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1D Potential
1. Homework Statement
Given:
U(x) = U_0 - U_n*e^-(kx^2), U_0, U_n and k are all constant.
i) What is the maximum E for the particle to remain in potential?
ii) Show that the potential has a minimum at x = 0.
iii) If a particle of mass m is placed near the minimum at x = 0, and displaced slightly, show that it does shm. find the period of this motion.2. Homework Equations
\frac{Df}{Dx} = -U(x)
3. The Attempt at a Solution
i) First, when I received this problem, it was in very bad handwriting. My immediate thought for this part of the problem was to differentiate U(x) with respect to x since I know the relevant equation from above, make the function negative, and set it equal to zero. However, I don't think Energy and potential are related like that, since as far as I know, the force has that relationship, not the energy - and I know very well that force and energy have different units =).
So my question for part i): Do you think I miswrote this problem? Does finding E-Max make sense given a potential?
ii) I'm not very comfortable with this problem. To find minima, I differentiate and set equal to zero. Differentiating U(x), we get
U(x) = -2xkU_n * e^(-kx^2)
And to find minima, we set U(x) to zero and solve for x.
0 = 2xkU_n, x = 0.
Then, I plug in values of -1 and 1 into the original equation, as well as zero, and zero clearly gives the lowest value (the x^2 ensures that!)
Is that sufficient to prove that there is a minimum at x = 0?
Finally,
iii) If a particle of mass m is placed near the minimum at x = 0, and displaced slightly, show that it does simple harmonic motion. Find the period of this motion.
Now this one, I'm a bit stumped on. While I don't know the graph of this motion, I do know that at x = 0 there is a minima, so it looks kind of like the minima in a parabola or something. (I believe this particle, at x = 0, is at 'stable' equilibrium, but I might be confusing terms...) treating this minima as a kind of 'ditch' that the particle cannot escape from without sufficient potential to overcome this 'ditch', the particle, starting at, say, -1, will start racing down, pass the ditch and, ignoring friction, make it to +1 before changing direction, passing x = 0 and moving back to -1. (oscillating!)
Clearly, this is SHM, and I know that for a SHM, F = -kx. I don't really know how to show this mathematically though. I was planning on rearranging the potential formula so that it looked something like this:
U(x) = U_0 - U_n*e^-kx^2,
U_0 - U(x) = U_n*e^-kx^2, Let [U_0 - U(x)]/(U_n) = Q,<br /> <br /> Q = e^-kx^2,<br /> <br /> LN(Q) = -kx^2, ...
But this feels definitely wrong.
If anyone can help me out on this, I'd be very appreciative. Thanks for taking your time to help me =).
1. Homework Statement
Given:
U(x) = U_0 - U_n*e^-(kx^2), U_0, U_n and k are all constant.
i) What is the maximum E for the particle to remain in potential?
ii) Show that the potential has a minimum at x = 0.
iii) If a particle of mass m is placed near the minimum at x = 0, and displaced slightly, show that it does shm. find the period of this motion.2. Homework Equations
\frac{Df}{Dx} = -U(x)
3. The Attempt at a Solution
i) First, when I received this problem, it was in very bad handwriting. My immediate thought for this part of the problem was to differentiate U(x) with respect to x since I know the relevant equation from above, make the function negative, and set it equal to zero. However, I don't think Energy and potential are related like that, since as far as I know, the force has that relationship, not the energy - and I know very well that force and energy have different units =).
So my question for part i): Do you think I miswrote this problem? Does finding E-Max make sense given a potential?
ii) I'm not very comfortable with this problem. To find minima, I differentiate and set equal to zero. Differentiating U(x), we get
U(x) = -2xkU_n * e^(-kx^2)
And to find minima, we set U(x) to zero and solve for x.
0 = 2xkU_n, x = 0.
Then, I plug in values of -1 and 1 into the original equation, as well as zero, and zero clearly gives the lowest value (the x^2 ensures that!)
Is that sufficient to prove that there is a minimum at x = 0?
Finally,
iii) If a particle of mass m is placed near the minimum at x = 0, and displaced slightly, show that it does simple harmonic motion. Find the period of this motion.
Now this one, I'm a bit stumped on. While I don't know the graph of this motion, I do know that at x = 0 there is a minima, so it looks kind of like the minima in a parabola or something. (I believe this particle, at x = 0, is at 'stable' equilibrium, but I might be confusing terms...) treating this minima as a kind of 'ditch' that the particle cannot escape from without sufficient potential to overcome this 'ditch', the particle, starting at, say, -1, will start racing down, pass the ditch and, ignoring friction, make it to +1 before changing direction, passing x = 0 and moving back to -1. (oscillating!)
Clearly, this is SHM, and I know that for a SHM, F = -kx. I don't really know how to show this mathematically though. I was planning on rearranging the potential formula so that it looked something like this:
U(x) = U_0 - U_n*e^-kx^2,
U_0 - U(x) = U_n*e^-kx^2, Let [U_0 - U(x)]/(U_n) = Q,<br /> <br /> Q = e^-kx^2,<br /> <br /> LN(Q) = -kx^2, ...
But this feels definitely wrong.
If anyone can help me out on this, I'd be very appreciative. Thanks for taking your time to help me =).