in damped oscillation, the amplitude of oscillation is reduced to one fourth of its initial value. 10 cm at the end of 50 oscillations. what will be its amplitude when the oscillator completes 150 oscillations?
To determine the amplitude of the oscillator when it completes 150 oscillations, we first need to find the damping factor.
The damping factor is the ratio of the amplitude at the end of one complete cycle to the amplitude at the start of the next cycle. In this case, the amplitude reduces to one fourth of its initial value after 50 oscillations.
Let's denote the initial amplitude as A₀. After 50 oscillations, the amplitude becomes A = A₀/4.
Now we can set up a formula to calculate the damping factor:
Damping factor (d) = A/A₀ = (A₀/4)/A₀ = 1/4
So, the damping factor is 1/4.
To find the amplitude after 150 oscillations, we can use the formula for damped oscillation:
Amplitude (Aₙ) = A₀ * (d^n)
Where Aₙ is the amplitude after n oscillations.
Substituting the values we know:
Aₙ = 10 cm * (1/4)^150
Now, we can calculate the amplitude after 150 oscillations:
Aₙ = 10 cm * (1/4)^150
Using a calculator, we find that Aₙ is approximately 0.000003 times the initial amplitude.
Therefore, the amplitude of the oscillator when it completes 150 oscillations will be almost zero.