
Simple Harmonic Motion  
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Posted by: pavan_609 ^{®} 05/06/2006, 01:40:58 Author Profile eMail author Edit 
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 Author Profile eMail author Edit 
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 Author Profile eMail author Edit 
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  zekeman  Post Reply  Top of thread  Forum 
Posted by: swearingen ^{®} 05/09/2006, 12:53:16 Author Profile eMail author Edit 
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 Author Profile eMail author Edit 
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 Author Profile eMail author Edit 
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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