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Deriving Equation - HSN forum

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Deriving Equation


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#1 Ian

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Posted 11 October 2005 - 07:03 PM

I would greatly appreciate it if someone could answer this question

Write an expression for the radial acceleration of a particle moving in a circle of radius r, with a speed of v

Hence show that the centrapertal force F acting on this particle of mass m is inversly proportional to the square of the period T

Thanks a lot

Ian

#2 dfx

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Posted 11 October 2005 - 08:58 PM

a = v power2.gif / r

F = ma = mv power2.gif / r

BUT v = rw and w = theta.gif/T = 2 pi.gif /T . So v = r 2 pi.gif / T

Then F = ( m 4 pi.gif power2.gif r power2.gif ) / T power2.gif r

F = ( m 4 pi.gif power2.gif r )/ T power2.gif

therefore.gif F is inversely proportional to T power2.gif

#3 dfx

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Posted 11 October 2005 - 09:01 PM

However, I have a question.

We know that F = ( mv power2.gif ) / r .

Say instead of v we substitute r.

So we know that V = ( 2 pi.gif r ) / T implies.gif r = vT/2 pi.gif

So F = ( m v power2.gif ) / r = ( m v power2.gif 2 pi.gif ) / vT = ( m v 2 pi.gif ) / T

So is this not contradictory to our previous conclusion? In this case F is inversely proportional to T only. huh.gif

#4 Ian

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Posted 11 October 2005 - 09:49 PM

Thanks a lot

#5 werlop

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Posted 12 October 2005 - 09:45 AM

I know this has been answered already, but I'll just put it in Tex for ease of reading.


\[
\begin{array}{l}
 a = \frac{{v^2 }}{r} \\ 
 F = \frac{{mv^2 }}{r}{\rm  } \\ 
 {\rm but}\;v = r\omega {\rm ,}\;v^2  = r^2 \omega ^2 {\rm   so }\;\underline {F = mr\omega ^2 }  \\ 
 \omega  = \frac{{2\pi }}{T}{\rm ,}\;\omega ^2  = \frac{{4\pi ^2 }}{{T^2 }} \\ 
 F = mr\omega ^2  = \frac{{4mr\pi ^2 }}{{T^2 }} \\ 
  \Rightarrow F = \frac{1}{{T^2 }} \times 4mr\pi ^2  \\ 
 \underline{\underline {\;F\;{\rm is}\;{\rm inversely}\;{\rm proportional}\;{\rm to}\;T^2 }}  \\ 
 \end{array}
\]
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Or being hated, don't give way to hating,
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#6 werlop

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Posted 12 October 2005 - 09:52 AM

QUOTE(dfx @ Oct 11 2005, 10:01 PM)
However, I have a question.

We know that F = ( mv power2.gif ) / r .

Say instead of v we substitute r.

So we know that V = ( 2 pi.gif r ) / T implies.gif r = vT/2 pi.gif

So F = ( m v power2.gif ) / r = ( m v power2.gif 2 pi.gif ) / vT =  ( m v 2 pi.gif ) / T

So is this not contradictory to our previous conclusion? In this case F is inversely proportional to T only.  huh.gif

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I did this really quickly so may have made a mistake, but it still works even if you substitute r:


\[
\begin{array}{l}
 a = \frac{{v^2 }}{r} \\ 
 F = \frac{{mv^2 }}{r} \\ 
 r = \frac{v}{\omega } \\ 
 {\rm but}\;v = r\omega {\rm ,}\;v^2  = r^2 \omega ^2 {\rm   so }\;\underline {F = mr\omega ^2 }  \\ 
 F = mr\omega ^2  = r\omega (m\omega ) = \underline {mv\omega }  \\ 
 \omega  = \frac{{2\pi }}{T}{\rm ,}\;v = \frac{{2\pi r}}{T} \\ 
 F = mv\omega  = m \times \frac{{2\pi r}}{T} \times \frac{{2\pi }}{T} \\ 
 F = \frac{{4mr\pi ^2 }}{{T^2 }} \\ 
  \Rightarrow F = \frac{1}{{T^2 }} \times 4mr\pi ^2  \\ 
 \underline{\underline {\;F\;{\rm is}\;{\rm inversely}\;{\rm proportional}\;{\rm to}\;T^2 }}  \\ 
 \end{array}
\]
Click here to visit the Bearsden Academy Website
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If you can keep your head when all about you
Are losing theirs and blaming it on you,
If you can trust yourself when all men doubt you
But make allowance for their doubting too,
If you can wait and not be tired by waiting,
Or being lied about, don't deal in lies,
Or being hated, don't give way to hating,
And yet don't look too good, nor talk too wise:
Yours is the Earth and everything that's in it,
And--which is more--you'll be a Man, my son!





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