13. Damped oscillator. A spherical ball of radius 0.30 cm and mass 0.5 g moves in water under

step 1 Here for the solution for the step 1 mass of the oscillator m equal to 0 .375 kg for constant k

13. Damped oscillator. A spherical ball of radius 0.30 cm...

13. Damped oscillator. A spherical ball of radius $0.30 \mathrm{~cm}$ and mass $0.5 \mathrm{~g}$ moves in water under the action of a spring of constant $C=50 \mathrm{dyn} / \mathrm{cm} . \eta$ for water is $1.0 \times 10^{-2} \mathrm{dyn}-\mathrm{s} / \mathrm{cm}^{2}$ or poises. Find the number of oscillations that will occur in the time for the amplitude to drop to one-half the initial amplitude. (Note that $e^{-0.693}=\frac{1}{2}$.) What is the $Q$ of the oscillator?

13. Damped oscillator. A spherical ball of radius $0.30 \mathrm{~cm}$ and mass $0.5 \mathrm{~g}$ moves in water under the action of a spring of constant $C=50 \mathrm{dyn} / \mathrm{cm} . \eta$ for water is $1.0 \times 10^{-2} \mathrm{dyn}-\mathrm{s} / \mathrm{cm}^{2}$
or poises. Find the number of oscillations that will occur in the time for the amplitude to drop to one-half the initial amplitude. (Note that $e^{-0.693}=\frac{1}{2}$.) What is the $Q$ of the oscillator?

Key Concepts

Damped Harmonic Oscillator

A damped harmonic oscillator is a system where a mass attached to a spring oscillates while experiencing a resistive force that is typically proportional to its velocity. This damping force causes the oscillations to gradually decrease in amplitude over time. Understanding this model is key for analyzing systems where energy is gradually lost, such as mechanical and electrical resonators.

Exponential Decay of Amplitude

Exponential decay describes how the amplitude of oscillations in a damped oscillator decreases over time according to an exponential function. This behavior is characterized by a decay constant that relates to the damping coefficient; the amplitude reduces to a fraction (like half) of its initial value over a specific time period, providing insight into the rate at which energy is dissipated.

Quality Factor (Q)

The Quality Factor, or Q, is a dimensionless metric that quantifies the damping of an oscillator. It represents the ratio of the stored energy to the energy lost per cycle and is inversely related to the rate of damping. A high Q indicates that the system loses energy slowly and oscillates for many cycles before the effects of damping become significant.

Damping Coefficient

The damping coefficient is a parameter that measures the magnitude of the resistive force acting on an oscillating system per unit velocity. It directly affects the rate at which energy is dissipated from the system, thereby influencing the exponential decay of amplitude and the overall dynamic response of the oscillator.

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