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Mechanics, PE to position function

  • Thread starter tjkubo
  • Start date
  • #1
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Homework Statement


A 3 kg object is moving along the x-axis where U(x) = 4x2. At x = -.5, v = +2. Find the object's position and KE as functions of time. Assume x = 0 at time t = 0. All forces acting on the object are conservative.

Homework Equations


ME = U + K
K = (1/2)mv2
F = dU/dx
F = ma

The Attempt at a Solution


Using initial conditions, ME = 4.
F = dU/dx = 8x
F = ma
8x = (3)d2x/dt2
This is where I got stuck. I was attempting to solve for x(t), find v(t), then use that to find K(t). Assuming everything else is correct, how do you solve a second order differential equation like this? Otherwise, please correct me.
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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Just solve

[tex]\frac{d^2x}{dt^2}- \frac{8}{3}x=0[/tex]

Do you know how to solve a second order differential equation with constant coefficients?


EDIT: http://www.sosmath.com/diffeq/second/constantcof/constantcof.html" [Broken]
 
Last edited by a moderator:
  • #3
42
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I am still having trouble solving for x(t).
I got [tex]x=c_1e^{\sqrt{\frac{8}{3}}t}+c_2e^{-\sqrt{\frac{8}{3}}t}[/tex]
and [tex]c_1+c_2=0[/tex]
but since there is no initial value associating time and velocity, I can't find the constants.
 
  • #4
rock.freak667
Homework Helper
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31
I am still having trouble solving for x(t).
I got [tex]x=c_1e^{\sqrt{\frac{8}{3}}t}+c_2e^{-\sqrt{\frac{8}{3}}t}[/tex]
and [tex]c_1+c_2=0[/tex]
but since there is no initial value associating time and velocity, I can't find the constants.
How did you get ME=4 by chance?

Also F=-dU/dx not F=+dU/dx
 
  • #5
Delphi51
Homework Helper
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10
Can you use the "At x = -.5, v = +2" condition?
Would it work to begin with
ME = U + K
4 = 4x^2 + 1/2*mv^2 (which includes the x = -.5 condition)
4 = 4x^2 + 1.5(dx/dt)^2
The solution to this differential equation would have only one constant, which you should be able to get using the x=0 at t=0 condition.
 
  • #6
42
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How did you get ME=4 by chance?

Also F=-dU/dx not F=+dU/dx
My bad, ME = 7. I forgot to square. (Is is correct to assume that ME is constant?)
Anyway, when I retried solving the differential equation with the initial conditions, I ended up getting 0 = 0 while solving for the constants. ????
 
  • #7
42
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4 = 4x^2 + 1.5(dx/dt)^2
The solution to this differential equation would have only one constant, which you should be able to get using the x=0 at t=0 condition.
How would you solve this differential equation? The (dx/dt)^2 term throws me off.
 
  • #8
Delphi51
Homework Helper
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10
dx/dt = sqrt(2/3)*sqrt(4 - 4x^2)
sqrt(2/3) dt = dx/sqrt(4 - 4x^2)
Integrate both sides. Doesn't look bad - trig substitution if I'm not mistaken.
 
  • #9
42
0
Yes! I got
[tex]x=\frac{\sqrt{7}}{2}\sin{\sqrt{\frac{4}{3}}\,t}[/tex]
That was much simpler than what I was doing.
Thanks Delphi!
 

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