A wooden rectangular prism of density \(ho_{w}\), height \(h\), and cross section \(a \times b\) is initially depressed in an oil tub and made to vibrate freely in the vertical direction (see Fig. 2.111). Use Rayleigh's method to find the natural frequency of vibration of the prism. Assume the density of oil is \(ho_{0}\). If the rectangular prism is replaced by a uniform circular cylinder of radius \(r\), height \(h\), and density \(ho_{w}\), will there be any change in the natural frequency?

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X a h FIGURE 2.111 Wooden prism vibrating in oil tub.