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Q. 72

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A oscillator in a vacuum chamber has a frequency of . When air is admitted, the oscillation decreases to of its initial amplitude in . How many oscillations will have been completed when the amplitude is of its initial value?

When the amplitude is of its initial value, the completed oscillation is

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Step by Step Solution

Step 1 Introduction

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

Step 2 Concepts and Principles

where denotes the angular frequency of the damped oscillator, is given by

The angular frequency of an undamped oscillator is .

Step 3 Concepts and Principles 

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

Given data:

The oscillator has a mass of:

The oscillator's frequency of oscillation is:

The oscillator's amplitude reduces to of its starting value over a time interval of .

Step 4 Required data and Solution

When the oscillator's amplitude drops to of its starting value, we must calculate the number of oscillations completed.

solution:

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

Step 5

where is the oscillator's initial amplitude. Rearrange the following and solve for :

Step 6

We substitute for because the oscillator's amplitude decreases to of its starting value at time :

Substitute the following numerical values:

Step 7

Solving Equation for yields the time it takes for the oscillator's amplitude to drop to of its starting value:

Step 8

We substitute for because we are looking for the period when the amplitude reduces to of its original values:

Substitute the following numerical values:

Step 9

Equation is used to calculate the oscillator's period of oscillation:

Substitute the following numerical values:

Step 10

The oscillator's total number of oscillations in time is then calculated.

oscillations

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