Conical Pendulum Part 3 Further Analysis with Calculated Results of the Period, Forces, Apex Angle, Pendulum Speed and Rotational Angular Momentum

Abstract

The conical pendulum provides a rich source of theoretical and computational analysis and the present work presents a seamless continuation of the previous publication. The tension force FT and centripetal force FC are explored further in linearization analyses and the appropriate slopes are explained. A similar analysis is applied to the period as a visual confirmation of the expected linearity and the slope is explained. The apex angle is considered as a function of both the conical pendulum period and angular speed. An analysis is presented of the conditions required for the constancy of the length of the adjacent side of the triangle defining the conical pendulum, which gives rise to an apparently counter-intuitive result and this is explained in detail. The speed of the rotating conical pendulum mass is calculated as a function of the radius and also as a function of the apex angle. The rotational speed is then used in order to calculate the rotational angular momentum. In order to maintain continuity of calculations from the previous work, the length range of the conical pendulum was maintained to be 0.435 m to 2.130 m, the local acceleration due to gravity g = 9.789 ms-2 and with a mass m = 0.1111 kg. This required the same limits for the string tension of approximately mg less than FT less than 12 N, the calculations therefore covered an apex angular range from zero (string hanging vertically down) up to 85°.