State the equation of motion and boundary conditions for torsional motion of an elastic rod that is fixed at on end and is attached to a rigid disk of mass moment of inertia \(I_{D}\) at its free end.

To determine the equation of motion for torsional motion of an elastic rod, we will use the basic equation considering the torque acting on it, given by: \(\tau = -GJ\frac{d\theta}{dx}\), where \(\tau\) is the torque, \(G\) is the shear modulus, \(J\) is the polar moment of inertia of the rod cross-section, \(\theta\) is the angular displacement, and \(x\) is the distance along the rod.

The torque acting on an elastic rod attached to a disk with mass moment of inertia \(I_{D}\) can be described as: \(\tau(x) = I_{D}\frac{d^{2}\theta(x)}{dt^{2}}\).

Now we will combine the basic torsional motion equation from Step 1 and the torque definition from Step 2: \( -GJ\frac{d\theta}{dx} = I_{D}\frac{d^{2}\theta(x)}{dt^{2}}\).

By rearranging the equation from Step 3, we can find the differential equation for the torsional motion of the elastic rod: \(\frac{d^{2}\theta(x)}{dx^{2}} = -\frac{GJ}{I_{D}}\frac{d^{2}\theta(x)}{dt^{2}}\).

In order to solve the equation of motion, we need to define the boundary conditions for the problem: Boundary Condition 1: At the fixed end of the elastic rod (i.e., \(x=0\)), the angular displacement is zero: \(\theta(0) = 0\). Boundary Condition 2: During free torsional motion of the disk, the torque is zero at the free end (i.e., \(x=L\)), where \(L\) represents the length of the elastic rod: \(\frac{d\theta(L)}{dx} = 0\). The equation of motion for the torsional motion of the elastic rod is given by the following differential equation: \(\frac{d^{2}\theta(x)}{dx^{2}} = -\frac{GJ}{I_{D}}\frac{d^{2}\theta(x)}{dt^{2}}\). And the boundary conditions are: 1. \(\theta(0) = 0\) 2. \(\frac{d\theta(L)}{dx} = 0\).