The Equation Of A Damped Harmonic Oscillation Is Given By 8(d2y/dt2) +24(dx/dt) + 48y =0. Find The Frequency Of The Damped Oscillations.

The Equation Of A Damped Harmonic Oscillation Is Given By :-

8(d2y/dt2) +24(dx/dt) + 48y =0.

Find The Frequency Of The Damped Oscillations

Ans) Method 1:-

The Equation Of A Damped Harmonic Oscillation Is Given By 8(d2y/dt2) +24(dx/dt) + 48y =0. Find The Frequency Of The Damped Oscillations.

Here's how to find the frequency of the damped oscillations from the given equation:

1. Identify the coefficients:

The coefficient of the second derivative term (8) represents the mass (m).

The coefficient of the first derivative term (24) represents the damping coefficient (b).

The coefficient of the position term (48) represents the spring constant (k).

2. Calculate the natural frequency:

The natural frequency (ω_0), which is the frequency of the system without damping, is given by:

ω_0 = √(k/m) = √(48/8) = √6 ≈ 2.45 rad/s.

3. Calculate the damping ratio:

The damping ratio (ζ), which characterizes the degree of damping, is given by:

ζ = b/(2√(mk)) = 24/(2√(8*48)) = 1/√2 ≈ 0.707

4. Determine the actual frequency of the damped oscillations:

Since the damping ratio is less than 1 (ζ < 1), the system is underdamped, meaning it will oscillate with a decreasing amplitude.

The frequency of the damped oscillations (ω_d) is given by:

ω_d = ω_0 √(1 - ζ^2) ≈ 2.45 √(1 - 0.707^2) ≈ 1.73 rad/s

Therefore, the frequency of the damped oscillations is approximately 1.73 rad/s.

Method 2 :-

The Equation Of A Damped Harmonic Oscillation Is Given By 8(d2y/dt2) +24(dx/dt) + 48y =
0. Find The Frequency Of The Damped Oscillations.

The Equation Of A Damped Harmonic Oscillation Is Given By 8(d2y/dt2) +24(dx/dt) + 48y =0. Find The Frequency Of The Damped Oscillations.

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किन्हीं दो विषयों पर टिप्पणी लिखिए:- १. टिप्पण की विशेषताएं २. प्रतिवेदन का महत्व ३. सामान्य हिंदी और कार्यालयी हिंदी ४. व्यावसायिक पत्र के प्रकार Kinhi Do Vishayon Par Tippan Likhiye | Hindi Bhasha 'Kha' तिप्पणी की विशेषताएँ और सुझाव: एक अच्छी टिप्पणी हमेशा सार्थक और संरचित होती है जो पाठकों को समझने में मदद करती है और सही दिशा में मार्गदर्शन करती है। यदि आप एक टिप्पणी लिख रहे हैं, तो इसे ध्यानपूर्वक सोचें और निम्नलिखित विशेषताओं का ध्यान रखें: सटीकता और संवेदनशीलता: आपकी टिप्पणी सटीक और संवेदनशील होनी चाहिए। जिससे पाठक आपकी बातों को सही से समझ सकें। संक्षेप में स्पष्टता: अधिक शब्दों की जगह, संक्षेप में स्पष्ट भाषा का उपयोग करें। सकारात्मक और सहायक: टिप्पणी सकारात्मक होनी चाहिए और पाठकों को कुछ सीखने के लिए प्रेरित करनी चाहिए। विवेचना करें: आपकी टिप्पणी में यदि विवेचना है, तो इसे और बढ़ावा देने के लिए समर्थन दीजिए। सही त्रुटियों को सुधारें: अगर आपने टिप्पणी में किसी त्रुटि को पहचाना है, तो उसे सही करें ताकि पाठकों को सही जानकारी मिले। इन सुझावों...

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