physics : class11 Velocity-Time Periodicity : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Velocity-Time Periodicity
MCQ questions and answers.
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MCQ Questions
1. A particle has velocity v(t) = 6 sin(4πt + π/3) m/s, where t is in seconds. What is the smallest positive time interval after which its velocity repeats exactly?
A.
0.125 s
B.
0.25 s
C.
0.50 s
D.
1.00 s
Correct Answer:
0.50 s
Explanation:
The angular frequency is 4π rad/s. The velocity period is T = 2π/ω = 2π/(4π) = 0.50 s. The phase shift does not affect the period.
2. The velocity of a particle is v(t) = 3 cos(5πt) + 4 cos(10πt) m/s. What is the fundamental period of the velocity-time function?
A.
0.20 s
B.
0.40 s
C.
0.60 s
D.
1.00 s
Correct Answer:
0.40 s
Explanation:
The individual periods are 2π/(5π) = 0.40 s and 2π/(10π) = 0.20 s. The smallest common repeating interval is 0.40 s, so this is the fundamental velocity period.
3. A velocity-time function is given by v(t) = 2 sin(πt/2) + 5 cos(πt/3) m/s. What is the smallest positive period of v(t)?
A.
6 s
B.
12 s
C.
18 s
D.
24 s
Correct Answer:
12 s
Explanation:
The component periods are T1 = 2π/(π/2) = 4 s and T2 = 2π/(π/3) = 6 s. The smallest common multiple of 4 s and 6 s is 12 s, which is therefore the period of the total velocity.
4. A particle's velocity is v(t) = 7 sin(ωt) + 7 cos(ωt). If the velocity repeats every 0.80 s, what is the value of ω?
A.
2.50π rad/s
B.
π/2 rad/s
C.
5π/2 rad/s
D.
5π rad/s
Correct Answer:
5π/2 rad/s
Explanation:
Both terms have angular frequency ω, so their sum has period T = 2π/ω. Thus ω = 2π/0.80 = 2.5π rad/s = 5π/2 rad/s.
5. The velocity of a particle is v(t) = 4 sin(3πt) m/s. At what earliest positive time does the velocity become identical to its value at t = 0, including the direction of motion?
A.
1/12 s
B.
1/6 s
C.
1/3 s
D.
2/3 s
Correct Answer:
2/3 s
Explanation:
The velocity period is T = 2π/(3π) = 2/3 s. Although the velocity has the same magnitude at earlier times, its exact signed value and direction repeat only after one complete period.
6. A velocity-time function has the form v(t) = A sin(2πt/T + φ). If v(0) = A/2 and the velocity is increasing at t = 0, which value of φ is possible, and what is the corresponding period when the angular frequency is 8π rad/s?
A.
φ = π/6, T = 0.25 s
B.
φ = π/6, T = 0.50 s
C.
φ = 5π/6, T = 0.25 s
D.
φ = 5π/6, T = 0.50 s
Correct Answer:
φ = π/6, T = 0.25 s
Explanation:
Since sin φ = 1/2, φ can be π/6 or 5π/6. The velocity is increasing when dv/dt = Aω cos φ is positive, which selects φ = π/6. The period is T = 2π/(8π) = 0.25 s.
7. A particle has a velocity function v(t) = 2 cos(4πt) - 3 sin(4πt) m/s. Which statement correctly describes its velocity-time periodicity?
A.
Its period is 0.25 s because the two terms have different phases.
B.
Its period is 0.50 s because both terms have angular frequency 4π rad/s.
C.
Its period is 1.00 s because the amplitudes are unequal.
D.
It is not periodic because sine and cosine are different functions.
Correct Answer:
Its period is 0.50 s because both terms have angular frequency 4π rad/s.
Explanation:
Sine and cosine terms with the same angular frequency combine into a single sinusoidal function with that same frequency. Therefore T = 2π/(4π) = 0.50 s.
8. The velocity of a particle is periodic and satisfies v(t + 1.2 s) = v(t) for every t. It also satisfies v(t + 0.8 s) = v(t) for every t. If the function is not constant, which could be its fundamental period?
A.
0.2 s
B.
0.4 s
C.
0.6 s
D.
0.8 s
Correct Answer:
0.4 s
Explanation:
The fundamental period must divide both 1.2 s and 0.8 s. Their greatest common divisor is 0.4 s, so 0.4 s is a possible fundamental period, whereas the other listed values do not satisfy both repetitions as the smallest period.
9. A velocity function is v(t) = 5 sin(2πt/3) + 5 sin(2πt/6) m/s. What is the smallest positive time τ for which v(t + τ) = v(t) for all t?
A.
1.5 s
B.
3 s
C.
6 s
D.
12 s
Correct Answer:
6 s
Explanation:
The first term has period 3 s, while the second has period 6 s. The total velocity repeats only when both terms repeat simultaneously, which first occurs after 6 s.
10. The velocity of a particle is v(t) = 8 cos(πt/2 + π/4) m/s. At what earliest positive time does the velocity become equal to its initial value while the slope of the velocity-time graph also has the same value as initially?
A.
0.5 s
B.
1.0 s
C.
2.0 s
D.
4.0 s
Correct Answer:
4.0 s
Explanation:
For both the velocity and its slope to repeat, the phase must increase by 2π. With angular frequency π/2 rad/s, the required time is τ = 2π/(π/2) = 4.0 s, the complete velocity period.
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