A simple pendulum of length L having a bob of mass m is...

A simple pendulum of length $L$ having a bob of mass $m$ is deflected from its rest position by an angle $\theta$ and released (figure 8-E16). The string hits a peg which is fixed at a distance $x$ below the point of suspension and the bob starts going in a circle centred at the peg. (a) Assuming that initially the bob has a height less than the peg, show that the maximum height reached by the bob equals its initial height. (b) If the pendulum is released with $\theta=90^{\circ}$ and $x=L / 2$ find the maximum height reached by the bob above its lowest position before the string becomes slack. (c) Find the minimum value of $x / L$ for which the bob goes in a complete circle about the peg when the pendulum is released from $\theta=90^{\circ} .$

Key Concepts

Conservation of Mechanical Energy

This concept explains that in the absence of non-conservative forces (like friction), the total mechanical energy (kinetic plus potential energy) of a system remains constant. In the context of pendulum motion, energy is converted between gravitational potential energy and kinetic energy. Analyzing these energy transformations allows one to determine quantities such as speed at various positions and maximum attainable heights.

Circular Motion and Centripetal Force

For an object undergoing circular motion, there must be a net inward (centripetal) force to keep it moving along a circular path. This force is provided by the tension in the string (or other constraint) and is calculated as the mass times the square of the velocity divided by the radius of the motion. Understanding the conditions required for maintaining circular motion, such as the minimum speed at the highest point of the circle, is crucial in such problems.

Pendulum Dynamics with a Variable Radius

This concept involves analyzing the motion of a pendulum when its effective length changes during its swing. When an external constraint (like a peg) alters the pendulum’s path, the system becomes segmented into parts with different radii and forces acting on it. Understanding how the geometry of the system affects energy conservation and dynamics is essential for solving problems where the pivot or center of rotation changes.

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