$(1)$ At the instant $\backslash (\backslash$theta$=60^\{\backslash$circ$\}\backslash ),$ for the bascule bridge, the cylinder AB is shortening with $\backslash ($ a$=0\mathrm{.}25\backslash )\backslash (\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ and $\backslash ($ v$=-0\mathrm{.}15\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Determine the angular velocity and angular acceleration the girder of the bridge.

Hint: Set the coordinate s$,$ horizontal distance between $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ),$ and relate $\backslash ($ s $\backslash )$ to the angular position $\backslash (\backslash$theta $\backslash ).$ Then, take time derivative of the position equation to find $\backslash ($ v$,\backslash$omega$,$ a $\backslash ),$ and $\backslash (\backslash$alpha $\backslash )$ relationships

MATLAB GUIDE:

$1.$ Relate $\backslash ($ s $\backslash )$ to theta.

$2.$ Define symbols for theta, s$,$ v$,$ a as functions of time and omega and alpha as variables using syms function in Matlab.

$3.$ Write the function of $\backslash ($ s $\backslash )$ with respect to theta.

$4.$ Use the diff command to write equations for v and a $.$

$5.$ Use subs command to substitute $\backslash (\backslash$frac$\{$d $\backslash$theta$\}\{$d t$\}\backslash )$ with omega on the equation that defined v $.$

$6.$ Use subs command to substitute $\backslash (\backslash$frac$\{$d$^\{2\}\backslash$theta$\}\{$d t$^\{2\}\}\backslash )$ with alpha on the equation that defined a$.$

$7.$ Use $\backslash ($ s u b s $\backslash )$ command to substitute $\backslash (\backslash$frac$\{$d $\backslash$theta$\}\{$d t$\}\backslash )$ with omega on the equation that defined a$.$

$8.$ Input the initial conditions of $\backslash (\backslash$mathrm$\{$v$\},\backslash$mathrm$\{$a$\}\backslash )$ and theta.

$9.$ Use solve command to substitute initial velocity in the v equation and solve the equation for omega.

$10.$ Use subs command to substitute initial value of theta in the omega equation $($from step $9)$

$11.$ Use $\backslash (\backslash$boldsymbol$\{$v p a$\}\backslash$boldsymbol$\{$a$\}\backslash )$ command to get the numerical value for omega.

$12.$ Use solve command to substitute initial acceleration in the a equation and solve the equation for alpha.

$13.$ Use subs command to substitute the calculated value of omega $($step $11)$ in the alpha equation $($from step $12)$

$14.$ Use subs command to substitute initial value of theta in the alpha equation $($from step $13)$

$15.$ Use $\backslash ($ v p a $\backslash )$ command to get the numerical value for alpha.

$16.$ Write the output and verify with the analytical answers for the question.

$17.$ Now put the condition of theta$=20$ degrees with the same velocity and acceleration of the cylinder and write the output as well.

$18.$ For the assignment, submit your code, the equations for $\backslash ($ v $\backslash ),$ a$,$ omega and alpha before numerical value output and these two numerical outputs. You can use any other similar programming language for results. You can also use any other ways to work in Matlab provided it shows the equation output as well. This is one of the ways to do it$.$