Question

Question: Does a pendulum of lengthoscillate faster on the earth or on the moon?

1. Take$l=3$ and for the acceleration of gravity on earth. Use a numerical solver to generate a numerical solution curve for the non linear model (6) subject to the initial condition$\theta \left(0\right)=1,{\theta }^{\text{'}}\left(0\right)=2$ . Repeat using the same values but use $0.165g$for the acceleration of gravity on the moon.
2. From the graphs in part (a), determine which pendulum oscillates faster. Which pendulum has the greater amplitude of motion?

Short answer

The earth's pendulum oscillates faster. The amplitude of the pendulum on the moon is greater.

Step-by-step solution

Generating the graph for the given situation

The given values are $l=3$and $g=32$ for the gravityacceleration on Earth. The acceleration for gravity on the moon is $0.165g$ . Let $x_1=\theta$and $x_2={\theta }^{\text{'}}$. Then (1) is equivalent to

$$\begin{gathered} \left\{{x^{\text{'}}}_1=x_2 \\ {x^{\text{'}}}_2=-\frac{g}{l}\sin \theta \right\} \end{gathered}$$

By using the above, we will plot the solution of the equation (1) with the given initial condition$\theta \left(0\right)=1,{\theta }^{\text{'}}\left(0\right)=2$,

The code for GNU Octave (or MATLAB) is given below,

Coding for the equation in MATLAB

From the above graph, we can come to an conclusion that the earth's pendulum oscillates faster. The amplitude of the pendulum on the moon is greater.

The coding for the above graph is as below,

$$\begin{gathered} f\_e=@\left(t,y\right)\left[y\left(2\right):-\left(\raisebox{1ex}{$32$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\times \sin \left(y\left(1\right)\right)\right]; \\ f\_e=@\left(t,y\right)\left[y\left(2\right):-\left(0.165\times \raisebox{1ex}{$32$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\times \sin \left(y\left(1\right)\right)\right]; \\ \left[tr\_e,y\_e\right]=ode45\left(f\_m,\left[0,20\right],\left[t,2\right]\right); \\ figure; \\ plot\left(tr\_e,y\_e\left(:,1\right),\text{'}r\text{'},tr\_m,y\_m\left(:,1\right),\text{'}b\text{'}\right); \\ legend\left(\text{'}Earth\text{'},\text{'}Moon\text{'}\right); \\ grdon; \\ title\left(\sin \left(theta\right)\text{'}\right); \end{gathered}$$

The pendulum on the Earth oscillates faster. The pendulum on the moon has the greater amplitude.