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A Damped Harmonic Oscillator Has The Amplitude Of 20cm. It Reduces To 2cm After 100 Oscillations Each Of The Time Period 4.6s. Calculate Its Logarithmic Damping Constant. Compute The Number Of Oscillations In Which The Amplitude Drops By 50%

Q) A Damped Harmonic Oscillator Has The Amplitude Of 20cm. It Reduces To 2cm After 100 Oscillations Each Of The Time Period 4.6s. Calculate Its Logarithmic Damping Constant. Compute The Number Of Oscillations In Which The Amplitude Drops By 50%.

A Damped Harmonic Oscillator Has The Amplitude Of 20cm. It Reduces To 2cm After 100 Oscillations Each Of The Time Period 4.6s. Calculate Its Logarithmic Damping Constant. Compute The Number Of Oscillations In Which The Amplitude Drops By 50%

Damped Harmonic Oscillator Calculations

Here's how we can calculate the logarithmic damping constant and the number of oscillations for a 50% drop in amplitude for the given damped harmonic oscillator:


1. Logarithmic Damping Constant:


The logarithmic damping constant (δ) represents the rate of decay of the amplitude per oscillation. We can calculate it using the formula:


δ = (1 / N) * ln(A₀ / A_f)

where:


N is the number of oscillations (100 in this case)

A₀ is the initial amplitude (20 cm)

A_f is the final amplitude after N oscillations (2 cm)

Plugging in the values, we get:


δ = (1 / 100) * ln(20 / 2) ≈ 0.02303


Therefore, the logarithmic damping constant is approximately 0.02303.

2. Number of Oscillations for 50% Drop:


We want to find the number of oscillations (N_50) required for the amplitude to drop by 50%, meaning the final amplitude is half of the initial amplitude (A_f = A₀/2). Using the same formula as before:

N_50 = (1 / δ) * ln(A₀ / (A₀/2))

N_50 = (1 / 0.02303) * ln(2) ≈ 30.103

Therefore, the amplitude drops by 50% after approximately 30.1 oscillations.


Summary:

The logarithmic damping constant of the oscillator is approximately 0.02303.

The amplitude drops by 50% after approximately 30.1 oscillations.

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