A solid cone of semi-vertical angle \( \alpha=30^{\circ} \) and height ' \( h \) ' is suspended vertically from its vertex as shown in the figure. The cone is slightly rotated from its vertical position and then released. If the time period of small oscillations of the cone is \( 2 \pi \sqrt{ k \left(\frac{13 h }{45 g }\right)} . \) Find the value of \( k \).

Question

A solid cone of semi-vertical angle \( \alpha=30^{\circ} \) and height ' \( h \) ' is suspended vertically from its vertex as shown in the figure. The cone is slightly rotated from its vertical position and then released. If the time period of small oscillations of the cone is \( 2 \pi \sqrt{ k \left(\frac{13 h }{45 g }\right)} . \) Find the value of \( k \).