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Title: Pendulum sets with different initial angle plus dampingPost by: Fu-Kwun Hwang on December 03, 2010, 09:08:38 am
This simulation show 11 pendulums with different initial angle(from
minimum angle to minimum angle+angle range)It can be run in two different modes: 1. $\frac{d^2\theta}{dt}=-\frac{\ell}{g}\sin\theta-b\frac{d\theta}{dt}$ (equation for real pendulum) 2. $\frac{d^2\theta}{dt}\approx-\frac{\ell}{g}\theta-b\frac{d\theta}{dt}$ (equation for small angle approximation :assume $\sin\theta\approx\theta$) The damping factor b=0 is the default. You can change it and find out what will happened. For the second mode: the period and damping rate are the same for all pendulums. Do you know why? Title: Re: Pendulum sets with different initial angle plus dampingPost by: Fu-Kwun Hwang on December 15, 2010, 07:47:40 pm
For a mass m attached to a spring with spring constant k
The force is proportional to the displacement$x$. $\vec{F}=-k*x$ Assume there is a damping force which is proportional to the velocity $F=m\frac{d^2m}{dt^2}=-k*x-b*v$ The above equation can be rewrite as $m\frac{d^2m}{dt^2}+b \frac{dx}{dt}+k*x=0$ Do you notice the similarity between equation for spring and pendulum set? Title: Re: Pendulum sets with different initial angle plus dampingPost by: lookang on December 17, 2010, 09:33:04 am
This simulation show 11 pendulums with different initial angle(from this mode is demonstrate real pendulum swings at different period T for different magnitude of angle ϑ.minimum angle to minimum angle+angle range)1. $\frac{d^2\theta}{dt}=-\frac{\ell}{g}\sin\theta-b\frac{d\theta}{dt}$ (equation for real pendulum) well done! i like the view of all pendulum all superposition together.(http://www.phy.ntnu.edu.tw/ntnujava/index.php?action=dlattach;topic=2010.0;attach=3645;image) 2. $\frac{d^2\theta}{dt}\approx-\frac{\ell}{g}\theta-b\frac{d\theta}{dt}$ (equation for small angle approximation :assume $\sin\theta\approx\theta$) For the second mode: the period and damping factor are the same for all pendulums. Do you know why? This is due to the small angle assumption say ϑ = 5 degree, the period T has to be the same for the all ϑ = 5 degree. i don't understand the question on "damping factor are the same for all pendulums", if b = 0.5, it will be the same of all models of the pendulums because that is the way the model is made to obey this mathematical equation $\frac{d^2\theta}{dt}\approx-\frac{\ell}{g}\theta-b\frac{d\theta}{dt}$ (equation for small angle approximation :assume $\sin\theta\approx\theta$). is my interpretation of the question correct? Title: Re: Pendulum sets with different initial angle plus dampingPost by: Fu-Kwun Hwang on December 17, 2010, 10:38:49 am
Why the angle $\theta$ has to be smaller than 5 degree to be able to satisfy the small angle approximation?
Why 5.5 can not be a small angle? Does 5 a magic number??? Quote i don't understand the question on "damping factor are the same for all pendulums" What I mean is "The period and damping rate are the same for all pendulums."Title: Re: Pendulum sets with different initial angle plus dampingPost by: lookang on December 17, 2010, 11:02:55 am
Why the angle $\theta$ has to be smaller than 5 degree to be able to satisfy the small angle approximation? Why 5.5 can not be a small angle? Does 5 a magic number??? students can key in 5/180*PI = 0.08726 in radian mode, students can key in sin(5/180*PI) = 0.08715 the students can realise the approximation is accurate for 2 decimal place only. students can key in 5.5/180*PI = 0.09599 in radian mode, students can key in sin(5.5/180*PI) = 0.09584 the students can realize the approximation is accurate for 2 decimal place only. it is a balance between practically observably angle of oscillation and accuracy in approximation of the assumption of $\frac{d^2\theta}{dt}\approx-\frac{\ell}{g}\theta-b\frac{d\theta}{dt}$ i don't understand the question on "damping factor are the same for all pendulums" What I mean is "The period and damping rate are the same for all pendulums."ic damping rate. then my earlier answer is acceptable? made to obey this mathematical equation? Title: Re: Pendulum sets with different initial angle plus dampingPost by: Fu-Kwun Hwang on December 17, 2010, 11:49:42 am
$\sin\theta=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\frac{\theta^9}{9!}-\frac{\theta^{11}}{11!}+...$
The error due to approximation can be estimated as $\frac{\theta^3/3!}{\theta}=\frac{\theta^2}{6}$ For 5 degree, the error is about $\frac{(5\pi/180)^2}{6}=0.00127= 0.127%$ which is a very small error. For 10 degree, the error is about 4*0.127%= 0.50% For 20 degree, the error is about 16*0.127=2.0% What I want to point out is: 1. 5 degree is not a magic number: user should provide the precission required in order to determined the maximum angle for good approximation. 2. Even 20 degree only produce 2% of error which is normally smaller than experiment error when student perform real experiment. The solution for RLC circuit is: $Q(t)=Q_0 e^{-\alpha t}e^{i\omega t}$ where $\alpha=\frac{R}{2L}, \omega=\sqrt{\frac{1}{LC}}$ Because $\alpha$ is independent of charge or current: i.e. damping rate is the same for the same R/L value. You should be able to find out solution for pendulum(compare equations for RLC and pendumum set) |