Question

Question: Repeat the two parts of problem 23 this time using the linear model (7)

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 model that we need to look is

$\frac{d^2\theta }{d{t}^2}+\frac{g}{l}\theta =0\to \left(1\right)$

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

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

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

The difference is that in this case , both the pendulums oscillates faster and the pendulum on the moon has slightly smaller amplitude.

The coding for the above graph is as below,

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

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