The cranking device shown consists of a mass-spring system of stiffiness \(k\) and mass \(m\) that is pin-connected to a massless rod which, in turn, is pin- connected to a wheel at radius \(R\), as indicated. If the mass moment of inertia of the wheel about an axis through the hub is \(I_{O}\), determine the natural frequency of the system. (The spring is unstretched when connecting pin is directly over hub ' \(O\) '.)

In this step, we want to find the potential energy of the mass-spring system when it is displaced from its equilibrium position. The potential energy of a mass-spring system is given by the equation \(U = \frac{1}{2} k x^2\), where \(k\) is the stiffness constant, and \(x\) is the displacement from equilibrium. Since the connecting pin is directly above the hub 'O', the system is in equilibrium and the potential energy is 0. Let's call the displacement of the mass from the unstretched position as \(x\). Then, the potential energy of the mass-spring system is given by \(U = \frac{1}{2} k x^2\).

Now we need to find the kinetic energy of the system when the mass is in motion. The kinetic energy of an object is given as \(T = \frac{1}{2} m v^2\), where \(m\) is the mass and \(v\) is the velocity of the object. We can relate the linear velocity of the mass to the angular velocity of the wheel using the equation \(v = R \omega\), where \(\omega\) is the angular velocity and \(R\) is the radius of the wheel. Then the kinetic energy of the mass is \(T_m = \frac{1}{2} m (R \omega)^2 = \frac{1}{2} m R^2 \omega^2\). The kinetic energy of the wheel is given by \(T_w = \frac{1}{2} I_O \omega ^2\), where \(I_O\) is the mass moment of inertia of the wheel about an axis through the hub. The total kinetic energy of the system is the sum of the kinetic energy of the mass and the wheel, which is \(T = T_m + T_w = \frac{1}{2} m R^2 \omega^2 + \frac{1}{2} I_O \omega ^2\).

Now we will apply the principle of conservation of energy, which states that the total mechanical energy (sum of potential energy and kinetic energy) of a conservative system remains constant over time. In other words, \(E = U + T\), where \(E\) is the total mechanical energy. Using the expressions for the potential and kinetic energy we obtained in steps 1 and 2, the equation of motion can be written as: \(E = \frac{1}{2} k x^2 + \frac{1}{2} m R^2 \omega^2 + \frac{1}{2} I_O \omega ^2\)

The natural frequency, denoted as \(\omega_n\), can be found by analyzing the coefficients of the equation of motion. The equation can be rewritten as: \(\frac{1}{2} k x^2 = \frac{1}{2} m R^2 \omega^2 + \frac{1}{2} I_O \omega ^2 - E\) Simplifying the equation and solving for the natural frequency, we get: \( k x^2 = (m R^2 + I_O) \omega^2 - 2E \) \(\omega_n^2 = \frac{k x^2 + 2E}{m R^2 + I_O}\) Therefore the natural frequency is given by: \(\omega_n = \sqrt{\frac{k x^2 + 2E}{m R^2 + I_O}}\) Now we have a formula for the natural frequency of the system in terms of the given parameters: stiffness constant \(k\), mass \(m\), radius of the wheel \(R\), mass moment of inertia of the wheel \(I_O\), and the total mechanical energy \(E\).