gibson:teaching:spring-2015:math445:lab12
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gibson:teaching:spring-2015:math445:lab12 [2015/04/20 14:21] gibson created |
gibson:teaching:spring-2015:math445:lab12 [2015/04/22 13:17] (current) gibson |
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| - | ====== Math 445 lab 11: the pendulum ====== | + | ====== Math 445 lab 12: tick tock, the pendulum clock ====== |
| **Problem 1:** In class we developed a linear system of differential equations of the form $dx/dt = f(x) = Ax$ for the plane pendulum, where | **Problem 1:** In class we developed a linear system of differential equations of the form $dx/dt = f(x) = Ax$ for the plane pendulum, where | ||
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| We used the approximation $\sin \theta \approx \theta$ for small $\theta$, i.e. small amplitude oscillations. A chief result of this derivation was that the frequency of oscillation of the pendulum is given by $\omega = \sqrt{g/l}$, where $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. | We used the approximation $\sin \theta \approx \theta$ for small $\theta$, i.e. small amplitude oscillations. A chief result of this derivation was that the frequency of oscillation of the pendulum is given by $\omega = \sqrt{g/l}$, where $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. | ||
| - | However if $\theta$ gets large the approximation $\sin \theta \approx \theta$ is no longer valid. This lab problem is an investigation of how the frequency of oscillation of the pendulum changes when the amplitude of the oscillation is large. | + | However if $\theta$ gets large the approximation $\sin \theta \approx \theta$ is no longer valid. This lab problem is an investigation of how the frequency of oscillation of the pendulum changes when the amplitude of the oscillation is large. Specifically, you'll determine if a real physical pendulum clock runs faster or slower than the idealized linear clock, and by how much, depending on amplitude. |
| **(a)** Revise the derivation from class to develop a nonlinear system of differential equations $dx/dt= f(x)$ that is valid for large $\theta$. | **(a)** Revise the derivation from class to develop a nonlinear system of differential equations $dx/dt= f(x)$ that is valid for large $\theta$. | ||
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| - | for some value of $c$. Determine what the value of $c$ is from your graph in %%(c)%%. | + | for some value of $c$. Determine what the value of $c$ is from your graph in %%(c)%%. Do this by adding a curve of the above form to your plot from %%(c)%%, and adjusting the constant $c$ until the curve goes through the data. What is the value of $c$? What does this mean for the accuracy of a pendulum clock? Would a real pendulum clock run fast or slow compared to the idealized linear clock if the amplitude of oscillation is too big? |
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gibson/teaching/spring-2015/math445/lab12.1429564918.txt.gz · Last modified: 2015/04/20 14:21 by gibson
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