Simple pendulum
A simple harmonic motion on a simple pendulum \(\big(\)of length \((l)\) and mass \((m)\)\(\big)\) can be excited with an initial angular displacement \((\theta_0)\) (drag the point on the sliders). The influence of the acceleration due to gravity \((g)\) can be observed by changing its value corresponding to Earth, Moon or Mars.
Following quantities are indicated as outputs:
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Play buttons can be used for the following purposes:
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Equilibrium and peak amplitude positions of the mass can be seen by clicking "Show Positions" checkbox. These positions are labeled as "E" and "A" respectively and can be seen in the angular displacement curve (blue coloured) whenever the mass of the pendulum reaches these positions.
The linear displacement vector can be seen on the mass by clicking "Show Disp. vector" checkbox.
The linear displacement vector can be seen on the mass by clicking "Show Disp. vector" checkbox.
Variations of instantaneous angular velocity & angular acceleration with respect to time can be seen along with corresponding linear velocity and linear acceleration vectors on the mass. (Click the "Show Velocity" & "Show Accel." checkboxes)
Observations:
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Total energy of the system \((=\) Potential energy + Kinetic energy\()\) is indicated with green and saffron coloured bars. (Click the "Show Energies" checkbox to see these bars)
Observations:
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While deriving the frequency of motion of a simple pendulum, a smaller angle approximation is used for trigonometric ratios. Hence, the time period (or frequency) of the motion obtained by approximation is lesser than that of the actual. By clicking the "Actual motion" checkbox, the difference between the time period (blue-continuous and red-dotted curves) can be observed.
Equation of motion of a simple pendulum can be expressed as \(\ddot{\theta}+\dfrac{g}{l}\sin\theta=0\) \((1)\)
In case of small angular displacement, the trigonometric ratios can be approximated up to second order as follows:
\(\sin\theta=\theta\;;\;\;\;\;\cos\theta=1-\dfrac{\theta^2}{2}\) \((2)\) Hence, Eq.\((1)\) is reduced to, \(\ddot{\theta}+\dfrac{g}{l}\theta=0\) \((3)\) Taking simple harmonic motion of the form, \(\theta=\theta_0\cos(\omega t)\) \((\Rightarrow\) \(\dot{\theta}=-\theta_0\omega\sin(\omega t)\,\&\,\ddot{\theta}=-\theta_0\omega^2\cos(\omega t)\)\()\), the angular frequency can be obtained as, \(\omega=\sqrt{\dfrac{g}{l}}\) Therefore, approximate time period of the motion is \(T=\dfrac{2\pi}{\omega}=2\pi\sqrt{\dfrac{l}{g}}\) |
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For angular displacement up to \(163.1^\circ\), Carvalhaes and Suppes (2008) found the following solution (of Eq.\((1)\)) for the time period with an error percentage less than 1%, compared to the series solution obtained by Nelson (1986),
\(T=2\pi\sqrt{\dfrac{l}{g}}\dfrac{4}{\left(1+\sqrt{\cos\dfrac{\theta_0}{2}}\right)^2}\) Time period obtained using smaller angle approximation can be used for \(\theta_0\) up to \(22.9^\circ\) with an error percentage less than 1% compared to the series solution. |
With smaller angle approximation, energies are computed with following expressions,
Kinetic energy \((KE)=\dfrac{1}{2}mv^2=\dfrac{1}{2}ml^2\dot{\theta}^2\)
Potential energy \((PE)=mgh=mgl(1-\cos\theta)\approx\dfrac{1}{2}mgl\theta^2\)
Kinetic energy \((KE)=\dfrac{1}{2}mv^2=\dfrac{1}{2}ml^2\dot{\theta}^2\)
Potential energy \((PE)=mgh=mgl(1-\cos\theta)\approx\dfrac{1}{2}mgl\theta^2\)