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In the physics of a pendulum, particularly in simple harmonic motion, gravitational potential energy (GPE) is a key concept. It is the energy possessed by an object due to its position relative to the Earth. For a pendulum, this energy is at its maximum when the pendulum reaches its highest point in its swing. At this maximum angle, which is given as 45° from the vertical in our example, the rock attached to the pendulum string is raised to its highest vertical position. This height can be calculated using the geometry of the oscillation, specifically the length of the pendulum and the angle it makes to the vertical.The formula to find the vertical height (h) is:- - - \[ h = L - L \cos(\theta) \\]Here, \(L\) is the length of the pendulum and \(\theta\) is the angle. Using this formula, and knowing \(\theta = 45^{\circ}\), we find the height the rock is raised to during its swing, which allows us to compute the gravitational potential energy using \(PE = mgh\). This potential energy can then be converted into kinetic energy when the pendulum reaches the lowest point of its swing.

Kinetic energy (KE) is the energy that an object possesses due to its motion. In the case of our pendulum, when the pendulum rock passes through the vertical position during its swing, all the gravitational potential energy it had at the maximum point is converted into kinetic energy.This conversion is crucial in understanding pendulum motion. As the pendulum swings down from its highest point, it speeds up – exactly the point at which it has zero potential energy and maximum kinetic energy.The formula for kinetic energy is:- - - \[ KE = \frac{1}{2}mv^2 \\]Where \(m\) is the mass of the rock and \(v\) is its speed at the lowest point. By using the principle of conservation of energy, you can set the potential energy at the maximum height equal to the kinetic energy at the lowest point. This allows us to solve for the speed \(v\) as:- - - \[ v = \sqrt{2gh} \\]Plugging in the known values for \(g\) (the acceleration due to gravity, which is \(9.81 \, \mathrm{m/s^2}\)), \(h\), and \(m\), we find the rock's velocity as it swings through the vertical position.

In a pendulum system, several forces come into play during its motion. Understanding these forces is essential to solving problems related to pendulum dynamics.When the pendulum is at its extreme positions (maximum angular displacement), the forces acting on it include:- The gravitational force pulling it downward.- The tension in the string which has a component that balances the gravitational force at an angle.As the pendulum swings through the vertical, centripetal force becomes critical. This force keeps the object moving in a circular path. At this point, the tension in the string must overcome both the gravitational pull and provide enough force to sustain the pendulum's circular motion.In terms of calculations, at the angle of \(45^{\circ}\):- The centripetal force requirement reduces the tension because a part of gravitational force assists motion.- The tension includes both gravity’s pull on the mass (\(mg \cos(\theta)\)) and the centripetal component \(\left( \frac{mv^2}{L} \right)\).When the pendulum is vertical, the tension is at its maximum because:- It must hold the weight of the rock and provide the centripetal force required to keep moving in a circle.Thus, tension at the vertical position is:- - - \[ T = mg + \frac{mv^2}{L} \\]This comprehensive look into forces reveals how pendulum dynamics rely on both gravitational and tension forces to maintain the graceful swinging motion.