Here \$\omega_n\$ (frequency) is not given.
Then upon plotting the Root Locus, how will we find \$\small K\$ at \$\small\zeta = 0.5\$ if we're not using a graph sheet and want to find everything mathematically?
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\$\begingroup\$ Are you sure the required zeta values are 0 and 0.5? \$\endgroup\$ – Chu Dec 30 '20 at 20:05
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\$\begingroup\$ ω ≠ W. [filler] \$\endgroup\$ – Hearth Dec 30 '20 at 20:29
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\$\begingroup\$ For \$\zeta=0\$, this occurs when the radial line on the imaginary axis, or just find the gain as which the root locus intersects with the imaginary axis. For \$\zeta=0.5\$, you need to plot it in Matlab and search the intersection of the radial line with the root locus. In my code, this occurs at \$K=112\$ and for \$\zeta=0, K=443\$ \$\endgroup\$ – CroCo Dec 30 '20 at 23:43
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