Q. 71

Expert-verifiedFound in: Page 418

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

A air-track glider is attached to a spring with spring constant . The damping constant due to air resistance is . The glider is pulled out from equilibrium and released. How many oscillations will it make during the time in which the amplitude decays to of its initial value?

The amplitude decays to of its initial value is oscillations.

Damped Oscillations: During oscillations, the mechanical energy in a real oscillating system decreases because external forces, such as drag, inhibit the oscillations and convert mechanical energy to thermal energy. The real oscillator is then said to be damped, as is its motion. If the damping force is given by , where is the oscillator's velocity and is the damping constant, the oscillator's displacement is given by

where , the damped oscillator's angular frequency, is given by

The angular frequency of an undamped oscillator is .

A basic harmonic oscillator's period is proportional to its angular frequency of oscillation as follows:

Given data:

The glider has a mass of:

The spring constant is as follows:

The damping constant is as follows:

The glider's initial amplitude is:

The number of completed oscillations after which the glider's amplitude drops to of its original value is what we're supposed to figure out.

Solution:

As a function of time, the amplitude of the ball's oscillation is expressed as follows:

where is the oscillator's initial amplitude. Substitute for as follows:

solve for :

Substitute the following numerical values:

Before Equation, the glider's angular frequency of oscillation is determined:

Substitute the following numerical values:

The glider's period of oscillation is then calculated using the following equation:

Substitute the following numerical values:

The glider's number of oscillations completed in time is:

Oscillations

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