A 250 g air-track glider is attached to a spring with spring...

A $250 \mathrm{g}$ air-track glider is attached to a spring with spring constant $4.0 \mathrm{N} / \mathrm{m}$. The damping constant due to air resistance is $0.015 \mathrm{kg} / \mathrm{s} .$ The glider is pulled out $20 \mathrm{cm}$ from equilibrium and released. How many oscillations will it make during the time in which the amplitude decays to $e^{-1}$ of its initial value?

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Key Concepts

Oscillation Period Calculation

For a mass-spring system, the period of oscillation is the time taken for one complete cycle and is calculated using T = 2??(m/k), where m is the mass and k is the spring constant. This period is critical for determining the number of oscillations that occur within a given time span.

Damping Time Constant

The damping time constant is a parameter that quantifies the rate at which the amplitude of a damped oscillator decays. It is derived from the balance between mass and damping coefficient and indicates the time required for the amplitude to reduce to a set fraction (e.g., 1/e) of its original value.

Damped Harmonic Oscillator

This concept refers to an oscillatory system where a restoring force (often provided by a spring) and a damping force (typically proportional to velocity) work together to produce motion. The damping term causes the amplitude of oscillation to decrease over time, leading to a gradual loss of energy in the system.

Exponential Decay of Amplitude

In damped oscillatory systems, the amplitude of the oscillation decreases exponentially over time. This decay is characterized by an exponential factor where the amplitude becomes a fraction (such as one?eighth or e?¹) of its initial value over a characteristic time period determined by the damping constant and the mass of the system.

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