Damped & Forced Oscillations
Q. In a forced oscillation, what happens when the driving frequency matches the natural frequency?
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A.
The system oscillates with minimum amplitude
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B.
The system oscillates with maximum amplitude
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C.
The system stops oscillating
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D.
The system oscillates at a different frequency
Solution
When the driving frequency matches the natural frequency, resonance occurs, leading to maximum amplitude.
Correct Answer: B — The system oscillates with maximum amplitude
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Q. In a forced oscillation, what happens when the driving frequency matches the natural frequency of the system?
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A.
The system oscillates with minimum amplitude
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B.
The system oscillates with maximum amplitude
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C.
The system stops oscillating
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D.
The system oscillates at a different frequency
Solution
When the driving frequency matches the natural frequency, resonance occurs, leading to maximum amplitude.
Correct Answer: B — The system oscillates with maximum amplitude
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Q. In a forced oscillation, what is the effect of increasing the amplitude of the driving force?
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A.
Decreases the amplitude of oscillation
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B.
Increases the amplitude of oscillation
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C.
Has no effect on amplitude
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D.
Causes the system to stop oscillating
Solution
Increasing the amplitude of the driving force generally increases the amplitude of the forced oscillation.
Correct Answer: B — Increases the amplitude of oscillation
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Q. In a forced oscillation, what is the effect of resonance?
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A.
Amplitude decreases
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B.
Amplitude increases significantly
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C.
Frequency decreases
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D.
Phase difference becomes zero
Solution
At resonance, the driving frequency matches the natural frequency of the system, leading to a significant increase in amplitude.
Correct Answer: B — Amplitude increases significantly
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Q. In a forced oscillation, what is the term for the maximum amplitude achieved at resonance?
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A.
Resonance peak
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B.
Damping peak
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C.
Natural frequency
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D.
Driving frequency
Solution
The maximum amplitude achieved at resonance is referred to as the resonance peak.
Correct Answer: A — Resonance peak
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Q. In forced oscillations, what is the effect of increasing the amplitude of the driving force?
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A.
Decreases the amplitude of oscillation
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B.
Increases the amplitude of oscillation
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C.
Has no effect on amplitude
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D.
Causes the system to stop oscillating
Solution
Increasing the amplitude of the driving force generally increases the amplitude of the forced oscillation.
Correct Answer: B — Increases the amplitude of oscillation
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Q. In forced oscillations, what is the phase difference between the driving force and the displacement at resonance?
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A.
0 degrees
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B.
90 degrees
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C.
180 degrees
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D.
270 degrees
Solution
At resonance, the phase difference between the driving force and the displacement is 0 degrees, meaning they are in phase.
Correct Answer: A — 0 degrees
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Q. What happens to the frequency of a damped oscillator as damping increases?
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A.
Frequency increases
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B.
Frequency decreases
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C.
Frequency remains the same
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D.
Frequency becomes zero
Solution
As damping increases, the frequency of the damped oscillator decreases.
Correct Answer: B — Frequency decreases
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Q. What happens to the frequency of oscillation in a damped system compared to an undamped system?
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A.
It increases
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B.
It decreases
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C.
It remains the same
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D.
It becomes zero
Solution
The frequency of oscillation in a damped system is lower than that of an undamped system due to energy loss.
Correct Answer: B — It decreases
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Q. What is the condition for a system to be critically damped?
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A.
Damping coefficient equals zero
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B.
Damping coefficient is less than the natural frequency
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C.
Damping coefficient equals the square root of the product of mass and spring constant
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D.
Damping coefficient is greater than the natural frequency
Solution
A system is critically damped when the damping coefficient equals the square root of the product of mass and spring constant.
Correct Answer: C — Damping coefficient equals the square root of the product of mass and spring constant
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Q. What is the condition for critical damping in a damped harmonic oscillator?
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A.
Damping coefficient equals zero
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B.
Damping coefficient equals mass times natural frequency
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C.
Damping coefficient equals twice the mass times natural frequency
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D.
Damping coefficient is less than mass times natural frequency
Solution
Critical damping occurs when the damping coefficient equals twice the mass times the natural frequency of the system.
Correct Answer: C — Damping coefficient equals twice the mass times natural frequency
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Q. What is the condition for critical damping in a damped oscillator?
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A.
Damping coefficient equals zero
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B.
Damping coefficient equals mass times natural frequency
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C.
Damping coefficient is less than mass times natural frequency
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D.
Damping coefficient is greater than mass times natural frequency
Solution
Critical damping occurs when the damping coefficient equals the mass times the natural frequency.
Correct Answer: B — Damping coefficient equals mass times natural frequency
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Q. What is the damping ratio for critically damped oscillation?
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A.
Less than 1
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B.
Equal to 1
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C.
Greater than 1
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D.
Zero
Solution
A critically damped system has a damping ratio equal to 1, which allows it to return to equilibrium without oscillating.
Correct Answer: B — Equal to 1
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Q. What is the effect of damping on the amplitude of an oscillating system?
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A.
Amplitude increases with time
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B.
Amplitude remains constant
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C.
Amplitude decreases with time
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D.
Amplitude becomes zero instantly
Solution
Damping causes the amplitude of oscillations to decrease over time due to energy loss.
Correct Answer: C — Amplitude decreases with time
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Q. What is the effect of damping on the energy of an oscillating system?
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A.
Energy increases
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B.
Energy remains constant
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C.
Energy decreases over time
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D.
Energy oscillates
Solution
Damping causes the energy of the oscillating system to decrease over time due to energy loss.
Correct Answer: C — Energy decreases over time
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Q. What is the effect of increasing the damping coefficient on the amplitude of oscillation in a damped oscillator?
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A.
Increases amplitude
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B.
Decreases amplitude
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C.
No effect
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D.
Doubles amplitude
Solution
Increasing the damping coefficient decreases the amplitude of oscillation in a damped oscillator.
Correct Answer: B — Decreases amplitude
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Q. What is the equation for the displacement of a damped harmonic oscillator?
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A.
x(t) = A e^(-bt) cos(ωt)
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B.
x(t) = A e^(bt) cos(ωt)
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C.
x(t) = A cos(ωt)
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D.
x(t) = A e^(-bt) sin(ωt)
Solution
The displacement of a damped harmonic oscillator is given by x(t) = A e^(-bt) cos(ωt), where b is the damping coefficient.
Correct Answer: A — x(t) = A e^(-bt) cos(ωt)
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Q. What is the equation of motion for a damped harmonic oscillator?
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A.
m d²x/dt² + b dx/dt + kx = 0
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B.
m d²x/dt² + kx = 0
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C.
m d²x/dt² + b dx/dt = 0
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D.
m d²x/dt² + b dx/dt + kx = F(t)
Solution
The equation of motion for a damped harmonic oscillator is m d²x/dt² + b dx/dt + kx = 0.
Correct Answer: A — m d²x/dt² + b dx/dt + kx = 0
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Q. What is the general form of the equation for a damped harmonic oscillator?
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A.
x(t) = A cos(ωt)
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B.
x(t) = A e^(-bt) cos(ωt)
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C.
x(t) = A sin(ωt)
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D.
x(t) = A e^(bt) cos(ωt)
Solution
The equation x(t) = A e^(-bt) cos(ωt) describes the motion of a damped harmonic oscillator.
Correct Answer: B — x(t) = A e^(-bt) cos(ωt)
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Q. What is the general form of the equation of motion for a damped harmonic oscillator?
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A.
m d²x/dt² + b dx/dt + kx = 0
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B.
m d²x/dt² + kx = 0
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C.
m d²x/dt² + b dx/dt = 0
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D.
m d²x/dt² + b dx/dt + kx = F(t)
Solution
The equation of motion for a damped harmonic oscillator includes a damping term and is given by m d²x/dt² + b dx/dt + kx = 0.
Correct Answer: A — m d²x/dt² + b dx/dt + kx = 0
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Q. What is the general form of the equation of motion for a damped oscillator?
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A.
m d²x/dt² + b dx/dt + kx = 0
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B.
m d²x/dt² + kx = 0
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C.
m d²x/dt² + b dx/dt = 0
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D.
m d²x/dt² + b dx/dt + kx = F(t)
Solution
The equation of motion for a damped oscillator includes a damping term (b dx/dt) along with the restoring force (kx).
Correct Answer: A — m d²x/dt² + b dx/dt + kx = 0
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Q. What is the phase difference between the driving force and the displacement in a damped forced oscillator at resonance?
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A.
0°
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B.
90°
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C.
180°
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D.
270°
Solution
At resonance, the phase difference is 90°.
Correct Answer: B — 90°
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Q. What is the phase difference between the driving force and the displacement in a forced oscillation at resonance?
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A.
0 degrees
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B.
90 degrees
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C.
180 degrees
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D.
270 degrees
Solution
At resonance, the phase difference between the driving force and the displacement is 0 degrees.
Correct Answer: A — 0 degrees
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Q. What is the phase difference between the driving force and the displacement in a damped oscillator at resonance?
-
A.
0 degrees
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B.
90 degrees
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C.
180 degrees
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D.
270 degrees
Solution
At resonance, the phase difference between the driving force and the displacement is 180 degrees.
Correct Answer: C — 180 degrees
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Q. What is the relationship between the amplitude of a damped oscillator and time?
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A.
Exponential decay
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B.
Linear decay
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C.
Quadratic decay
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D.
Constant decay
Solution
The amplitude of a damped oscillator decreases exponentially with time.
Correct Answer: A — Exponential decay
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Q. What is the relationship between the damping coefficient and the type of damping?
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A.
Higher coefficient indicates under-damping
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B.
Lower coefficient indicates over-damping
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C.
Critical damping occurs at a specific coefficient
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D.
Damping coefficient has no effect
Solution
Critical damping occurs at a specific value of the damping coefficient, which separates under-damping from over-damping.
Correct Answer: C — Critical damping occurs at a specific coefficient
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Q. What is the relationship between the damping ratio and the type of damping in a system?
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A.
Damping ratio < 1 indicates overdamping
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B.
Damping ratio = 1 indicates critical damping
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C.
Damping ratio > 1 indicates underdamping
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D.
Damping ratio = 0 indicates critical damping
Solution
A damping ratio of 1 indicates critical damping, while less than 1 indicates underdamping and greater than 1 indicates overdamping.
Correct Answer: B — Damping ratio = 1 indicates critical damping
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Q. What is the relationship between the damping ratio and the type of damping?
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A.
Damping ratio < 1: Underdamping
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B.
Damping ratio = 1: Overdamping
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C.
Damping ratio > 1: Critical damping
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D.
Damping ratio = 0: Overdamping
Solution
A damping ratio less than 1 indicates underdamping, equal to 1 indicates critical damping, and greater than 1 indicates overdamping.
Correct Answer: A — Damping ratio < 1: Underdamping
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Q. What is the time period of a damped oscillator with a damping ratio of 0.1 and a natural frequency of 10 rad/s?
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A.
0.2 s
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B.
0.3 s
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C.
0.4 s
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D.
0.5 s
Solution
Time period (T) = 2π/ω_n = 2π/10 = 0.2π ≈ 0.628 s.
Correct Answer: C — 0.4 s
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Q. What is the time period of a damped oscillator with a natural frequency of 3 rad/s and a damping ratio of 0.1?
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A.
2π/3
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B.
2π/3.1
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C.
2π/3.2
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D.
2π/3.3
Solution
Time period (T) = 2π/ω_n = 2π/3 rad/s = 2π/3.
Correct Answer: A — 2π/3
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