A body of mass m is tied to one end of a string of length l...

A body of mass $m$ is tied to one end of a string of length $l$ and revolves vertically in a circular path. At the lowest point of circle, what must be the $K . E$. of the body so as to complete the circle (a) $5 \mathrm{mgl}$ (b) $4 \mathrm{mgl}$ (c) $2.5 \mathrm{mgl}$ (d) $2 \mathrm{mgl}$

A body of mass $m$ is tied to one end of a string of length $l$ and revolves vertically in a circular path. At the lowest point of circle, what must be the $K . E$. of the body so as to complete the circle
(a) $5 \mathrm{mgl}$
(b) $4 \mathrm{mgl}$
(c) $2.5 \mathrm{mgl}$
(d) $2 \mathrm{mgl}$

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy a body possesses due to its position in a gravitational field. In vertical circular motion, the difference in height between the lowest and highest points of the circle results in a change in GPE. This change must be taken into account when calculating the kinetic energy required at the lowest point, as some of the energy will be converted into potential energy as the body moves upward, affecting its ability to complete the circle.

Conservation of Energy

This concept involves equating the total mechanical energy at different points along the motion of a body. In problems involving vertical circular motion, energy conservation is used to relate the kinetic energy at the lowest point to the sum of kinetic and potential energies at higher points. It allows us to determine the minimum amount of kinetic energy required at the bottom so that even after converting some of it to potential energy at the top, the body can maintain the necessary speed to stay in circular motion.

Circular Motion Dynamics

Circular motion dynamics deals with the forces and accelerations a body experiences when moving in a circular path. Specifically, in vertical circular motion, there is an interplay between the gravitational force and the tension in the string, which must together provide the necessary centripetal force for the motion. Understanding these forces is crucial for determining conditions, such as the minimum speed or kinetic energy needed at various positions of the circle to ensure that the motion remains continuous and the string remains taut.

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