Numerical Integration Project

In this project you will have to analytically and numerically integrate the same acceleration equation and

A particle, having mass $m,$ is suspended by two cables $AB$ and $AC$ in a vertical plane. Then cable $AB$ is cut

and the ball begins to swing. The equations, in Normal$-$Tangential Polar Coordinates, for this problem

are:

${\displaystyle \underset{?}{\overset{?}{}}}{F}_t=mr{}^{}=-$mgsin$$

${\displaystyle \underset{?}{\overset{?}{}}}{F}_n=mr{}^{}{?}^2=T-$mgcos$$

where:

m : mass of the ball, $2\{kg\}$

r : length of the string, $0\mathrm{.}5\{m\}$

g: gravity $\{\frac{m}{s^2}\}$

$,{}^{},{}^{}$ : Angular position, velocity, and acceleration of the mass about point C in $\{rad\},\{ra\frac{d}{s}\},$ and

$\{ra\frac{d}{s^2}\}$ respectively

Determine analytically:

The angular velocity of the mass about point C as a function of angular position, ${}^{}(),$ by

integration of the equation of motion, ${}^{}.$ Initial angular position of $-\frac{}{6}$rad.

The tension in the string as a function of angular position, $T().$

Determine numerically:

The angular velocity and angular position of the mass about point $C$ as a function of time by

integration of the equation of motion, ${}^{}$

The tension in the string as a function of time.

Deliverables:

Plot both the analytical and numerical solutions of ${}^{}().$ Use a solid black line for the analytical

solution and a dashed red line for the numerical solution. Include appropriate labels and a

legend.

Plot the relative error between the numerical and analytical solutions:

$Re{l}_{\text{e}\text{r}\text{r}\text{o}\text{r}}=\frac{{}_{\text{n}\text{u}\text{m}\text{e}\text{r}\text{i}\text{c}\text{a}\text{l}}^{}-{}_{\text{a}\text{n}\text{a}\text{l}\text{t}\text{i}\text{c}\text{a}\text{l}}^{}}{{}_{\text{a}\text{n}\text{a}\text{l}\text{y}\text{t}\text{i}\text{c}\text{a}\text{l}}^{}}100\%$

How does this compare to the default relative tolerance of ode$45?$