A homogeneous rod AB, of mass m and length 2a, hangs in a thread from a fixed point O. The thread is unstretchable and massless and of length da. See figure. The distance AC is 2a/3. The rod performs small oscillations in a vertical plane containing O. Determine lambda such that AB perpetually forms an angle with the vertical twice as big as OC does, if the system is started suitably.Treat the rod as one-dimensional, and assume that it never collides with the thread.

Q1: The system shown has two masses. Beam of mass (Jo#m L² kg.m²) rotates about fixed point (O) and its free end is connected to disk rotates about fixed point (O₂). Consider all connecting links are massless and rigid. Find 1- The displacements of points A, B, and C in addition to the rotations of masses, all in terms of 0. 2- Find the equation of motion (EOM) in terms of 0. 3- What is the natural frequency of the system? L/2 A Jo=m L² k L/2 Joz-m R² full 0₂ R U B C

Vibration

A thin, circular ring of mass m and radius r is suspended at 0. Determine the natural frequency of small oscillations of this ring in the xy plan. y Hint: The moment of inertia of a thin, circular ring of mass m and radius r about its central axis is I = mr². In this problem, the ring rotates about point 0; therefore, we need to use the Parallel Axis Theorem.