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Simple Harmonic Motion
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Posted by: pavan_609
Barney
05/06/2006, 01:40:58

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Can anyone give me the analytical explaination on "what is the effect on time period of simple pendulum having a ball containing water and continuously draining from it"




PAVAN


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Re: Simple Harmonic Motion
Re: Simple Harmonic Motion -- pavan_609 Post Reply Top of thread Forum
Posted by: swearingen

05/08/2006, 09:19:38

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The period of a simple pendulum is not affected by mass when taking the normal assumptions for pendulums. Just look at the equation - there is no m term. It is only dependent on length of the cable and gravity. Therefore, the period will not change. The amplitude will change, however.







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Re: Re: Simple Harmonic Motion
Re: Re: Simple Harmonic Motion -- swearingen Post Reply Top of thread Forum
Posted by: zekeman

05/09/2006, 00:15:32

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You are incorrect.
If you look at the differential equation I posted, you would see that there is a term in m' which is zero if the mass remains constant and only then you have the standard linear diff eq. But the m' in this case is a real negative number and leads to the equation with the term m' clearly shown and is not the simple harmonic you suggest







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Re: Re: Re: Simple Harmonic Motion
Re: Re: Re: Simple Harmonic Motion -- zekeman Post Reply Top of thread Forum
Posted by: swearingen

05/09/2006, 12:53:16

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You are technically correct, however, using your own assumption of sine(x) = x is erroneous as well. Why do you do so? Because it has very little effect on the practical answer. Mathematically, yes the change in mass will matter, as will assuming sine(x) = x. But if you go out to a swing and time it (as I have - physics experiments with grade school and basic college physics students is a specialty of mine), you'll find that what I said holds true.







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Re: Re: Re: Re: Simple Harmonic Motion
Re: Re: Re: Re: Simple Harmonic Motion -- swearingen Post Reply Top of thread Forum
Posted by: zekeman

05/11/2006, 08:37:17

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The assumption of the sine(x) and x is indeed valid for small angles even up to about 30 degrees where
x=pi/6=0.523
sine(pi/6)=.5
0r an assumption error of 4.6% in the coefficient of term in x which is very reasonable; however, the term in x' in my equation,m'/m could be very substantial.
You could do an experiment with a substantially leaking pail of water at the end of a string to see this. Depending on the leak rate you might see a very different motion than a pure harmonic.







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Re: Simple Harmonic Motion
Re: Simple Harmonic Motion -- pavan_609 Post Reply Top of thread Forum
Posted by: zekeman

05/06/2006, 15:33:15

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You first write the Newtonian differential equation
d(ml^2x')/dt= -mglx
where x is the angular displacement after assuming that the sine(x) is equal to x, the usual assumption for a pendulum.
Expanding I get
x"-(m'/m)x'+(g/l)x=0
where m'is the rate of loss of mass
In general, you can't solve this linear equation in closed form but there are plenty of numerical techniques available to get a solution.Qualitatively, it should be noted that the negative sign of the x' coefficient indicates, in general, an unstable or rising amplitude solution of the basic harmonic.







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