A cylinder of mass \(m\) and mass moment of inertia \(J\) is connected to a spring of stiffness \(k\) and rolls on a rough surface as shown in Fig. 2.112. If the translational and angular displacements of the cylinder are \(x\) and \(\theta\) from its equilibrium position, determine the following:

a. Equation of motion of the system for small displacements in terms of \(x\) using the energy method.

b. Equation of motion of the system for small displacements in terms of \(\theta\) using the energy method.

c. Find the natural frequencies of the system using the equation of motion derived in parts

(a) and (b). Are the resulting natural frequencies same?

Transcribed Image Text:

Cylinder rolls without sliding FIGURE 2.112 Cylinder connected to a spring. Rough surface