A hot-air balloon of mass \(m\) is used to lift a load, \(M g\), by means of 12 equally spaced elastic ropes, each of stiffness \(k\) (see Fig. 5.60). Find the natural frequencies of vibration of the balloon in vertical direction. State the assumptions made in your solution and discuss their validity.

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Balloon, mass m 12 Elastic ropes, stiffness k each Weight (Mg) FIGURE 5.60 Hot-air balloon connected to a mass.