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Kinetic energy is the energy that an object possesses due to its motion. It depends on two factors: the mass of the object and its velocity. Mathematically, kinetic energy is expressed by the equation: \[ K.E. = \frac{1}{2}mv^2 \] where \( m \) is the mass and \( v \) is the velocity of the object.In the context of vertical circular motion, the kinetic energy is crucial at the lowest point of the circle. At this point, the velocity of the object must be high enough to ensure that it can overcome gravity and travel to the top. The solution shows how kinetic energy at this point must be sufficient to eventually equate to the combination of potential energy and the energy needed for circular motion at the top.When performing calculations that involve kinetic energy as seen in this exercise, remember to address both the mass and velocity. As the mass influences both kinetic and potential energy, understanding this relationship helps us comprehend energy transformations in motion.

Potential energy is the stored energy in an object due to its position relative to other objects. For an object in a gravitational field, like Earth, it is often associated with its height above the ground. The formula for potential energy is: \[ P.E. = mgh \] where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point.In this exercise, the body reaches the highest point of its motion in a vertical circle. Here, its height is maximum, thus providing it maximum potential energy. As calculated, in a vertical circle of radius \( l \), the height \( h \) at the top point is \( 2l \). Therefore, the potential energy at this point is \( 2mgl \).This part of the exercise highlights how potential energy at the highest point in a circular motion connects with kinetic energy to form the basis of energy conservation.

Vertical circular motion involves an object moving in a circle in a plane perpendicular to the force of gravity. This motion combines elements of both kinetic and potential energy, and it’s crucial to understand how both forms of energy interchange as the object moves.For an object to maintain vertical circular motion, conservation of energy principles dictate that the energy balance is crucial. The object must have sufficient kinetic energy at the lowest point to convert into potential energy (while also overcoming gravitational force) to reach the top of the circle.The calculated required kinetic energy at the bottom point is actually \( 3mgl \) when adhering to energy conservation laws and accounting for gravity’s role in the motion. This ensures that while it ascends, converting kinetic energy to potential energy, it maintains forward momentum. This is why understanding the principles of vertical circular motion can illuminate how energy conservation helps achieve physical motion goals.