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For an underdamped harmonic oscillator with velocity v(t),
- a)Rate of energy dissipation varies linearly with v(t)
- b)Rate of energy dissipation varies as square of v(t)
- c)The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)
- d)For weak damping, the amplitude decays exponentially to zero
Correct answer is option 'B,C,D'. Can you explain this answer?
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For an underdamped harmonic oscillator with velocity v(t),a)Rate of en...
Underdamped Harmonic Oscillator:
A harmonic oscillator is a system that exhibits periodic motion around an equilibrium position. In the case of an underdamped harmonic oscillator, there is some damping present in the system, which causes the amplitude of the oscillations to decrease over time.
Rate of Energy Dissipation:
The rate of energy dissipation in a system is determined by the damping factor. In the case of an underdamped harmonic oscillator, the rate of energy dissipation is not constant but depends on the velocity of the oscillator.
a) Rate of energy dissipation varies linearly with v(t)
This statement is incorrect. The rate of energy dissipation does not vary linearly with velocity in an underdamped harmonic oscillator. Instead, it varies as the square of the velocity.
b) Rate of energy dissipation varies as the square of v(t)
This statement is correct. The rate of energy dissipation in an underdamped harmonic oscillator varies as the square of the velocity. This can be understood by considering the equation of motion for the oscillator, which includes a damping term proportional to the velocity squared. This term represents the dissipation of energy due to damping.
c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)
This statement is correct. The reduction in the oscillator frequency, also known as the damping factor, is determined by the damping coefficient and the mass of the oscillator. It is independent of the velocity of the oscillator.
d) For weak damping, the amplitude decays exponentially to zero
This statement is correct. In the case of weak damping, the amplitude of the oscillations decays exponentially to zero. This can be understood by considering the solution to the equation of motion, which involves a decaying exponential term. As time goes on, the amplitude of the oscillations decreases until it becomes negligible.
In summary, the correct statements for an underdamped harmonic oscillator are:
b) Rate of energy dissipation varies as the square of v(t)
c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)
d) For weak damping, the amplitude decays exponentially to zero.
A harmonic oscillator is a system that exhibits periodic motion around an equilibrium position. In the case of an underdamped harmonic oscillator, there is some damping present in the system, which causes the amplitude of the oscillations to decrease over time.
Rate of Energy Dissipation:
The rate of energy dissipation in a system is determined by the damping factor. In the case of an underdamped harmonic oscillator, the rate of energy dissipation is not constant but depends on the velocity of the oscillator.
a) Rate of energy dissipation varies linearly with v(t)
This statement is incorrect. The rate of energy dissipation does not vary linearly with velocity in an underdamped harmonic oscillator. Instead, it varies as the square of the velocity.
b) Rate of energy dissipation varies as the square of v(t)
This statement is correct. The rate of energy dissipation in an underdamped harmonic oscillator varies as the square of the velocity. This can be understood by considering the equation of motion for the oscillator, which includes a damping term proportional to the velocity squared. This term represents the dissipation of energy due to damping.
c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)
This statement is correct. The reduction in the oscillator frequency, also known as the damping factor, is determined by the damping coefficient and the mass of the oscillator. It is independent of the velocity of the oscillator.
d) For weak damping, the amplitude decays exponentially to zero
This statement is correct. In the case of weak damping, the amplitude of the oscillations decays exponentially to zero. This can be understood by considering the solution to the equation of motion, which involves a decaying exponential term. As time goes on, the amplitude of the oscillations decreases until it becomes negligible.
In summary, the correct statements for an underdamped harmonic oscillator are:
b) Rate of energy dissipation varies as the square of v(t)
c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)
d) For weak damping, the amplitude decays exponentially to zero.
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For an underdamped harmonic oscillator with velocity v(t),a)Rate of energy dissipation varies linearly with v(t)b)Rate of energy dissipation varies as square of v(t)c)The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)d)For weak damping, the amplitude decays exponentially to zeroCorrect answer is option 'B,C,D'. Can you explain this answer?
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For an underdamped harmonic oscillator with velocity v(t),a)Rate of energy dissipation varies linearly with v(t)b)Rate of energy dissipation varies as square of v(t)c)The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)d)For weak damping, the amplitude decays exponentially to zeroCorrect answer is option 'B,C,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about For an underdamped harmonic oscillator with velocity v(t),a)Rate of energy dissipation varies linearly with v(t)b)Rate of energy dissipation varies as square of v(t)c)The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)d)For weak damping, the amplitude decays exponentially to zeroCorrect answer is option 'B,C,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For an underdamped harmonic oscillator with velocity v(t),a)Rate of energy dissipation varies linearly with v(t)b)Rate of energy dissipation varies as square of v(t)c)The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)d)For weak damping, the amplitude decays exponentially to zeroCorrect answer is option 'B,C,D'. Can you explain this answer?.
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