A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.0

step 1 Now for this question we're looking at a pendulum and as oscillating in a simple harmonic motion

A simple pendulum with a length of 2.23 m and a mass of 6.74...

A simple pendulum with a length of $2.23 \mathrm{m}$ and a mass of $6.74 \mathrm{kg}$ is given an initial speed of $2.06 \mathrm{m} / \mathrm{s}$ at its equilibrium position. Assume it undergoes simple harmonic motion. Determine (a) its period, (b) its total energy, and (c) its maximum angular displacement.

Key Concepts

Period of a Pendulum

The period of a pendulum is the time it takes for one complete cycle of its swing. For small oscillations, the period is independent of mass and is given by the formula T = 2??(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This relationship assumes the small-angle approximation for the displacement.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to displacement and acts in the opposite direction. In the context of a pendulum with small angular displacements, the motion can be approximated as SHM, allowing the use of simple formulas to determine its period and other characteristics.

Conservation of Energy

Conservation of energy in oscillatory systems refers to the idea that the total mechanical energy (sum of potential and kinetic energy) remains constant in the absence of non-conservative forces. In a pendulum, the kinetic energy at the lowest point and the gravitational potential energy at the highest point are interchanged, and this principle is used to calculate quantities such as total energy based on the initial velocity or displacement.

Maximum Angular Displacement

The maximum angular displacement (or amplitude) of a pendulum is the greatest angle reached from the vertical during its swing. This can be determined from energy conservation by equating the initial kinetic energy to the gravitational potential energy gained at maximum displacement, taking into account the pendulum's length and the small-angle approximation for simplicity in the calculation.

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