A physics student investigates the oscillations of a small bob attached to a light, inextensible string of length L. In separate trials the student releases the bob from rest at two different initial angular displacements: Trial 1 at 5mo and Trial 2 at 10mo. Air resistance is negligible and the string remains taut during the motion.
Assume the small-angle approximation is valid for both trials.
(a) Describe the condition under which the small-angle approximation is considered valid for a simple pendulum. Explain how that condition applies to the two initial angles used in the experiment.
(b) The student claims that, to first order, the time required for the bob to complete one full oscillation is the same in both trials. Provide a qualitative explanation supporting or refuting the student’s claim. Your reasoning should explicitly reference the relationship between the restoring torque and the angular displacement for small angles.
(c) If the initial angular displacement were tripled (still remaining within the range for which the small-angle approximation is valid), predict how—if at all—the period would change. Justify your answer without performing a numerical calculation.