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If the amplitude of a damped - oscillator becomes one - third in 2 minutes, what will be the amplitude of oscillation as compared to initial one after 6 minutes?
- a)1/27
- b)1/9
- c)1/3
- d)27
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If the amplitude of a damped - oscillator becomes one - third in 2 min...
In case of damped oscillations, amplitude decrease exponentially with time , i.e
So
So
Most Upvoted Answer
If the amplitude of a damped - oscillator becomes one - third in 2 min...
Given:
- The amplitude of a damped oscillator becomes one-third in 2 minutes.
To find:
- The amplitude of oscillation as compared to the initial one after 6 minutes.
Let's solve this step by step:
1. Understanding the Damped Oscillator:
A damped oscillator is a system where the amplitude of oscillation decreases over time due to the presence of damping forces. These forces can be friction, air resistance, or any other dissipative force.
2. Amplitude of a Damped Oscillator:
The amplitude of a damped oscillator can be expressed as:
A(t) = A0 * e^(-bt)
Where:
- A(t) is the amplitude at time t
- A0 is the initial amplitude
- b is the damping constant
3. Finding the Damping Constant:
To find the damping constant, we can use the given information that the amplitude becomes one-third in 2 minutes.
Let's substitute the values:
1/3 = A0 * e^(-2b)
Taking the natural logarithm on both sides:
ln(1/3) = ln(A0 * e^(-2b))
ln(1/3) = ln(A0) - 2b
4. Solving for the Damping Constant:
From the equation above, we can isolate the damping constant:
2b = ln(A0) - ln(1/3)
2b = ln(A0) + ln(3)
b = (ln(A0) + ln(3))/2
5. Finding the Amplitude after 6 minutes:
Now that we have the damping constant, we can find the amplitude after 6 minutes using the given equation:
A(t) = A0 * e^(-bt)
Substituting the values:
A(6) = A0 * e^(-(ln(A0) + ln(3))/2 * 6)
A(6) = A0 * e^(-3ln(A0) - 3ln(3))
Using the logarithmic identity e^(-a) = 1/e^a:
A(6) = A0 / e^(3ln(A0) + 3ln(3))
A(6) = A0 / (e^(ln(A0))^3 * e^(ln(3))^3)
A(6) = A0 / (A0^3 * 3^3)
A(6) = 1/(27 * A0)
Therefore, the amplitude of oscillation after 6 minutes as compared to the initial amplitude is 1/27. Hence, the correct answer is option A.
- The amplitude of a damped oscillator becomes one-third in 2 minutes.
To find:
- The amplitude of oscillation as compared to the initial one after 6 minutes.
Let's solve this step by step:
1. Understanding the Damped Oscillator:
A damped oscillator is a system where the amplitude of oscillation decreases over time due to the presence of damping forces. These forces can be friction, air resistance, or any other dissipative force.
2. Amplitude of a Damped Oscillator:
The amplitude of a damped oscillator can be expressed as:
A(t) = A0 * e^(-bt)
Where:
- A(t) is the amplitude at time t
- A0 is the initial amplitude
- b is the damping constant
3. Finding the Damping Constant:
To find the damping constant, we can use the given information that the amplitude becomes one-third in 2 minutes.
Let's substitute the values:
1/3 = A0 * e^(-2b)
Taking the natural logarithm on both sides:
ln(1/3) = ln(A0 * e^(-2b))
ln(1/3) = ln(A0) - 2b
4. Solving for the Damping Constant:
From the equation above, we can isolate the damping constant:
2b = ln(A0) - ln(1/3)
2b = ln(A0) + ln(3)
b = (ln(A0) + ln(3))/2
5. Finding the Amplitude after 6 minutes:
Now that we have the damping constant, we can find the amplitude after 6 minutes using the given equation:
A(t) = A0 * e^(-bt)
Substituting the values:
A(6) = A0 * e^(-(ln(A0) + ln(3))/2 * 6)
A(6) = A0 * e^(-3ln(A0) - 3ln(3))
Using the logarithmic identity e^(-a) = 1/e^a:
A(6) = A0 / e^(3ln(A0) + 3ln(3))
A(6) = A0 / (e^(ln(A0))^3 * e^(ln(3))^3)
A(6) = A0 / (A0^3 * 3^3)
A(6) = 1/(27 * A0)
Therefore, the amplitude of oscillation after 6 minutes as compared to the initial amplitude is 1/27. Hence, the correct answer is option A.
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If the amplitude of a damped - oscillator becomes one - third in 2 min...
In case of damped oscillations, amplitude decrease exponentially with time , i.e
So
So
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If the amplitude of a damped - oscillator becomes one - third in 2 minutes, what will be the amplitude of oscillation as compared to initial one after 6 minutes?a)1/27b)1/9c)1/3d)27Correct answer is option 'A'. Can you explain this answer?
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