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How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position?
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How long after the begining of motion is the displacement of harmonica...
Understanding Harmonic Motion
In harmonic motion, the displacement of a particle from its mean position can be described by the equation:
Displacement (x) = A * sin(ωt)
where A is the amplitude, ω is the angular frequency, and t is time.
Given Data
- Period (T) = 24 seconds
- Amplitude (A) = A (unknown but constant)
Finding Angular Frequency
Angular frequency (ω) is calculated using the formula:
ω = 2π / T
Substituting the given period:
- ω = 2π / 24 = π / 12 rad/s
Displacement at Half Amplitude
To find the time when the displacement equals half the amplitude:
- Set x = A / 2 in the displacement equation:
A / 2 = A * sin(ωt)
- Simplifying gives:
1/2 = sin(ωt)
Finding Time (t)
- The inverse sine function gives:
ωt = sin⁻¹(1/2)
- The value of sin⁻¹(1/2) is π/6 radians.
- Therefore:
ωt = π/6
Substituting for ω:
(π / 12)t = π / 6
- Solving for t:
t = (π / 6) * (12 / π) = 2 seconds
Conclusion
The displacement of the harmonically oscillating point is equal to one-half its amplitude 2 seconds after the beginning of motion.
In harmonic motion, the displacement of a particle from its mean position can be described by the equation:
Displacement (x) = A * sin(ωt)
where A is the amplitude, ω is the angular frequency, and t is time.
Given Data
- Period (T) = 24 seconds
- Amplitude (A) = A (unknown but constant)
Finding Angular Frequency
Angular frequency (ω) is calculated using the formula:
ω = 2π / T
Substituting the given period:
- ω = 2π / 24 = π / 12 rad/s
Displacement at Half Amplitude
To find the time when the displacement equals half the amplitude:
- Set x = A / 2 in the displacement equation:
A / 2 = A * sin(ωt)
- Simplifying gives:
1/2 = sin(ωt)
Finding Time (t)
- The inverse sine function gives:
ωt = sin⁻¹(1/2)
- The value of sin⁻¹(1/2) is π/6 radians.
- Therefore:
ωt = π/6
Substituting for ω:
(π / 12)t = π / 6
- Solving for t:
t = (π / 6) * (12 / π) = 2 seconds
Conclusion
The displacement of the harmonically oscillating point is equal to one-half its amplitude 2 seconds after the beginning of motion.
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How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position?
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How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position?.
How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How long after the begining of motion is the displacement of harmonically oscillating point equal to one half its amplitude if the period is 24 seconds and particle started from mean position?.
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