In a science museum, a 110 kg brass pendulum bob swings at...

In a science museum, a $110 \mathrm{kg}$ brass pendulum bob swings at the end of a 15.0 -m-long wire. The pendulum is started at exactly 8: 00 A.M. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only $0.010 \mathrm{kg} / \mathrm{s} .$ At exactly 12: 00 noon, how many oscillations will the pendulum have completed and what is its amplitude?

Key Concepts

Oscillation Count

The oscillation count is determined by dividing the total time duration of the motion by the period of oscillation. This concept helps in quantifying how many complete cycles an oscillator goes through over a particular time span, which is especially useful when tracking repetitive motions in time-dependent problems.

Time Period Calculation

The time period of a pendulum undergoing simple harmonic motion 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 formula is crucial for determining how long each oscillation lasts, which in turn helps in calculating the number of oscillations over a given time interval.

Exponential Decay of Amplitude

The amplitude in a damped oscillator decreases in an exponential manner, typically represented by an expression of the form A(t) = A?e^(-?t/2m), where A? is the initial amplitude, ? is the damping constant, and m is the mass. This decay represents the loss of energy due to non-conservative forces like friction or air resistance.

Simple Pendulum

A simple pendulum is a mass suspended from a fixed point by a string such that it swings due to gravity. In the ideal small-angle approximation, it exhibits periodic motion with a period that depends only on the length of the pendulum and the gravitational acceleration, making it a classic example of simple harmonic motion.

Damped Harmonic Motion

Damped harmonic motion refers to oscillatory motion in which a resistive force, proportional to the velocity, causes the amplitude of the oscillation to decrease over time. This damping effect results in an exponential decay in amplitude, while the oscillation frequency may be slightly altered depending on the strength of the damping.

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