(Solved): Please show all work neatly. Thank You A pendulum is suspended by a linear spring as shown in ...

Please show all work neatly. Thank You

A pendulum is suspended by a linear spring as shown in the figure. The equation of motion is given by \[ \begin{array}{l} m L^{2} \theta+k L^{2} \sin \theta \cos \theta-m g L \cos \theta=0 \\ \ddot{\theta}+\frac{k}{m} \sin \theta \cos \theta-\frac{g}{L} \cos \theta=0 \end{array} \] Let \( m=2.8 \mathrm{~kg}, L=1.0 \mathrm{~m}, k=98 \mathrm{~N} / \mathrm{m} \), and \( g=9.81 \mathrm{~m} / \mathrm{s}^{\wedge} 2 \). Find the equilibrium position between 0 and \( 90^{\circ} \), and linearize the equation of motion about this equilibrium position. The linearized equation of motion can be written in the form, \( \theta+a \theta=b \), where \( a \) and \( b \) are constant. What is a? Do not include units, and keep 3 significant figures. Write your answer in scientific notation (eg. \( 10=1.00 \mathrm{e} 1 \) ).