A square loop is made of four identical uniform solid cylinders, each of mass m , radius R , and length L . The loop is hinged freely along the entire length of its top side (the longitudinal axis of the top cylinder) so that it can oscillate as a physical pendulum in a vertical plane. The time period of small oscillations of this square loop is :

Step-by-step solution

To find the time period of the physical pendulum, we first calculate the moment of inertia of the square loop about the hinge axis (the longitudinal axis of the top cylinder). 1. Top cylinder (axis is its own longitudinal axis): I_ top = 1 2 m R² 2. Bottom cylinder (parallel to the hinge axis at a distance L ): Using the parallel axis theorem: I_ bottom = 1 2 m R² + m L² 3. Two side cylinders (axis is perpendicular to their length and passes through one end): For each side cylinder, I = I_ cm + m ( L 2 )² = ( m R²