Kinematics: find the particle's position as a function of time

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Homework Statement


There is a particle that moves in 1-dimension. mass is m, force on the object is given as a function of position(x), Fx= -mω2x.(ω is constant) when t=0 particle has x0>0, and velocity is zero. find the particle's position as a function of time.

Homework Equations


There is a given hint for the problem. Use z=Acosu conversion on this integral
nSdRc5I.jpg


The Attempt at a Solution


What i did so far: I used Newton's second law so that i can find accerelation. Then i integrate accerelation function and find velocity function as a function of position. Lastly i integrate velocity function and i find x= x02x3/6. From there i can't do anything. I couldn't use motion equations because accerelation is not constant. And i couldn't find how to use this integral.
 

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  • #2
haruspex
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What i did so far:
Please post your working, or we have little idea where you are going wrong.
(It looks like you have the wrong idea about how to solve differential eauations. You cannot integrate ∫f(x).dt as though it were ∫f(x).dx.)
 
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Please post your working, or we have little idea where you are going wrong.
(It looks like you have the wrong idea about how to solve differential eauations. You cannot integrate ∫f(x).dt as though it were ∫f(x).dx.)
Yes you are right. If i didn't make a mistake i find x= x02xt2/2
 
  • #4
haruspex
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Yes you are right. If i didn't make a mistake i find x= x02xt2/2
No, you can't do that either. That is treating x as a constant.
I don't uderstand why the hint quotes that integral, but the substitution it recommends should help. Try x=A cos(u).
 
  • #5
Orodruin
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I don't uderstand why the hint quotes that integral,
I do. It is the integral you get in order to solve the separable differential equation obtained if you first multiply Newton 2 with ##\dot x## and integrate it.
 
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