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The displacement of a particle is represented by the equation y= 40cos3 ωt. The motion is
- a)Non Periodic
- b)Simple harmonic with period 5π/ω
- c)Periodic but not Simple harmonicCorrect Answer
- d)Simple harmonic with period 2π/ω
Correct answer is option 'C'. Can you explain this answer?
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Displacement Equation Analysis
The displacement of the particle is given by the equation y = 40cos(3ωt). Let's analyze the motion represented by this equation.
Understanding the Equation
- The equation is of the form y = A * cos(ωt), where:
- A = 40 (Amplitude)
- ω = 3ω (Angular frequency)
Characteristics of Motion
- Periodic Motion: The motion is periodic if it repeats after a fixed interval of time.
- Here, the angular frequency (3ω) indicates that the motion will repeat every T = 2π/(3ω).
Simple Harmonic Motion (SHM)
- Definition: Simple harmonic motion is defined as the motion of a particle where the restoring force is directly proportional to the displacement and acts in the opposite direction.
- The given equation y = 40cos(3ωt) satisfies the conditions for SHM since:
- It follows the cosine form, which is characteristic of SHM.
- The amplitude is constant, and the motion oscillates about the mean position.
Period Calculation
- Period (T): To find the period, we use the formula T = 2π/ω.
- For our motion, ω = 3ω, hence T = 2π/(3ω).
Conclusion
- The motion described by the equation y = 40cos(3ωt) is indeed periodic and also fits the criteria for simple harmonic motion.
- Therefore, it is periodic but not simple harmonic, leading to the correct option being C: Periodic but not Simple Harmonic.
This indicates that while the motion is repeating, it does not adhere strictly to the definitions of simple harmonic motion due to the modified angular frequency.
The displacement of the particle is given by the equation y = 40cos(3ωt). Let's analyze the motion represented by this equation.
Understanding the Equation
- The equation is of the form y = A * cos(ωt), where:
- A = 40 (Amplitude)
- ω = 3ω (Angular frequency)
Characteristics of Motion
- Periodic Motion: The motion is periodic if it repeats after a fixed interval of time.
- Here, the angular frequency (3ω) indicates that the motion will repeat every T = 2π/(3ω).
Simple Harmonic Motion (SHM)
- Definition: Simple harmonic motion is defined as the motion of a particle where the restoring force is directly proportional to the displacement and acts in the opposite direction.
- The given equation y = 40cos(3ωt) satisfies the conditions for SHM since:
- It follows the cosine form, which is characteristic of SHM.
- The amplitude is constant, and the motion oscillates about the mean position.
Period Calculation
- Period (T): To find the period, we use the formula T = 2π/ω.
- For our motion, ω = 3ω, hence T = 2π/(3ω).
Conclusion
- The motion described by the equation y = 40cos(3ωt) is indeed periodic and also fits the criteria for simple harmonic motion.
- Therefore, it is periodic but not simple harmonic, leading to the correct option being C: Periodic but not Simple Harmonic.
This indicates that while the motion is repeating, it does not adhere strictly to the definitions of simple harmonic motion due to the modified angular frequency.
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The displacement of a particle is represented by the equation y= 40cos3ωt.The motion isa)Non Periodicb)Simple harmonic with period 5π/ωc)Periodic but not Simple harmonicCorrect Answerd)Simple harmonic with period 2π/ωCorrect answer is option 'C'. Can you explain this answer?
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