physics : class11 Acceleration-Time Periodicity : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Acceleration-Time Periodicity
MCQ questions and answers.
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MCQ Questions
1. A particle has acceleration a(t) = 3 cos(6t) + 4 sin(10t), where t is in seconds. What is the fundamental time period of the acceleration?
A.
π/5 s
B.
π/2 s
C.
π s
D.
2π s
Correct Answer:
π s
Explanation:
The angular frequencies are 6 rad s⁻¹ and 10 rad s⁻¹. Their greatest common divisor is 2 rad s⁻¹, so the fundamental period is T = 2π/2 = π s.
2. The acceleration of a particle is given by a(t) = A sin(4πt) + B cos(6πt), where A and B are nonzero constants. What is the smallest positive period of a(t)?
A.
1/6 s
B.
1/3 s
C.
1 s
D.
2 s
Correct Answer:
1 s
Explanation:
The individual periods are 1/2 s and 1/3 s. The smallest time that is an integral multiple of both is 1 s, so the combined acceleration repeats after 1 s.
3. A particle undergoes simple harmonic motion with displacement x = A cos(ωt + φ). Which statement about its acceleration-time periodicity is correct?
A.
The acceleration has period 2π/ω and phase 180° relative to displacement.
B.
The acceleration has period π/ω and phase φ relative to displacement.
C.
The acceleration has period 2π/ω and is in phase with velocity.
D.
The acceleration has period 4π/ω and phase 90° relative to displacement.
Correct Answer:
The acceleration has period 2π/ω and phase 180° relative to displacement.
Explanation:
For SHM, a = −ω²x = −ω²A cos(ωt + φ). The negative sign gives a phase difference of π, while the angular frequency and therefore the period remain unchanged at 2π/ω.
4. The acceleration of a particle is a(t) = a₀ cos²(ωt), where a₀ is nonzero. What is the fundamental period of the acceleration?
A.
2π/ω
B.
π/ω
C.
π/(2ω)
D.
4π/ω
Correct Answer:
π/ω
Explanation:
Using cos²(ωt) = [1 + cos(2ωt)]/2, the time-dependent part has angular frequency 2ω. Hence the acceleration repeats after T = 2π/(2ω) = π/ω.
5. A periodic acceleration is described by a(t) = a₁ cos(ωt) + a₂ cos(3ωt), with both coefficients nonzero. Which is its fundamental period?
A.
π/(3ω)
B.
π/ω
C.
2π/ω
D.
6π/ω
Correct Answer:
2π/ω
Explanation:
The periods of the two terms are 2π/ω and 2π/(3ω). The smallest common period is 2π/ω; the third-harmonic term also completes an integer number of cycles during this interval.
6. An acceleration-time graph repeats exactly after 0.40 s, but within each 0.40 s interval it has two identical cycles. What is the fundamental period of the acceleration?
A.
0.10 s
B.
0.20 s
C.
0.40 s
D.
0.80 s
Correct Answer:
0.20 s
Explanation:
If two identical cycles occur in 0.40 s, the smallest repeating interval is 0.40/2 = 0.20 s. The fundamental period is the smallest positive repetition time, not merely a larger graph interval.
7. The acceleration is a(t) = 5 sin(2πt/T) − 2 sin(6πt/T), where T is positive and both terms are present. What is the fundamental period of the acceleration?
A.
T/3
B.
T/2
C.
T
D.
3T
Correct Answer:
T
Explanation:
The first term has period T, while the second has period T/3. The smallest common period of both terms is T, so the acceleration-time function has fundamental period T.
8. A particle's acceleration is a(t) = a₀ sin(ωt + π/4). At t = 0, the acceleration is positive and increasing. Which conclusion about its periodicity is correct?
A.
Its period is π/ω because acceleration changes sign after half a cycle.
B.
Its period is 2π/ω, independent of the phase constant π/4.
C.
Its period is 4π/ω because the initial phase is π/4.
D.
Its period is π/(2ω) because the initial phase is π/4.
Correct Answer:
Its period is 2π/ω, independent of the phase constant π/4.
Explanation:
A phase shift changes where the acceleration cycle begins but not its repetition interval. Since the angular frequency is ω, the period remains 2π/ω.
9. The acceleration of a particle is a(t) = C[cos(ωt) + cos(2ωt) + cos(4ωt)], where C is nonzero. What is the fundamental period?
A.
π/(4ω)
B.
π/(2ω)
C.
π/ω
D.
2π/ω
Correct Answer:
2π/ω
Explanation:
The angular frequencies are ω, 2ω, and 4ω. The fundamental angular frequency is ω, so all terms simultaneously repeat after 2π/ω. The presence of higher harmonics does not reduce the period below that of the fundamental component.
10. An acceleration-time function satisfies a(t + 0.75 s) = a(t) for every t, and also satisfies a(t + 0.50 s) ≠ a(t) for at least one value of t. Which statement is necessarily true?
A.
Its fundamental period is 0.25 s.
B.
Its fundamental period is 0.50 s.
C.
Its fundamental period is 0.75 s.
D.
Its fundamental period is greater than 0.75 s.
Correct Answer:
Its fundamental period is 0.75 s.
Explanation:
The acceleration repeats after 0.75 s, so 0.75 s is a period. Since 0.50 s is not a period, and no smaller divisor is specified as a period, the stated repeating interval is the fundamental period under the given condition.
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