Daily Olympiad: Physics - Oscillations Waves [20260511] – 7 Practice Problems & Printable PDF

1. A particle executes SHM with a time period T. If its maximum velocity is v_max and maximum acceleration is a_max, what is the ratio of their amplitudes of acceleration to velocity?

A. a_max / v_max B. v_max * T C. 2πv_max / a_max D. a_max / (2πv_max)

Solution

Correct: A

In SHM, the maximum velocity is v_max = Aω and maximum acceleration is a_max = Aω². The ratio a_max/v_max = ω = 2π/T. Since ω = √(a_max/v_max²), the ratio of amplitudes (Aω/Aω²) gives 1/ω. Hence, the ratio is a_max / v_max.

2. Two wave functions y₁ = 4sin(2x - t) and y₂ = 4sin(2x + t) superpose. What is the amplitude of the resultant standing wave at x = λ/4?

A. 4√2 B. 8 C. 2√2 D. 0

Solution

Correct: B

Superposition gives y = 8cos(2x)sin(t). At x = λ/4, cos(2x) = cos(π) = -1. Amplitude is |8cos(2x)| = 8. Hence, the maximum amplitude is 8.

3. A closed pipe resonates with a tuning fork at its fundamental frequency. If the open end is suddenly converted to an open pipe, the new fundamental frequency becomes

A. Same as original B. Double the original C. Three times the original D. Four times the original

Solution

Correct: B

Fundamental frequency of closed pipe is f = v/(4L). For open pipe, f' = v/(2L). Hence, f' = 2f.

4. A wave pulse on a string reflects off a rigid boundary. The reflected pulse has

A. Inverted amplitude and same phase B. Inverted amplitude and opposite phase C. Same amplitude and same phase D. Same amplitude and opposite phase

Solution

Correct: B

Reflection from a rigid boundary inverts the wave due to Newton's third law (equal and opposite force). Hence, the phase changes by π.

5. A spring-mass system oscillates horizontally with amplitude A. When it reaches x = A/2, its potential energy is E_p. What is its kinetic energy at x = -A/2?

A. E_p B. 2E_p C. E_p/2 D. 3E_p/4

Solution

Correct: A

Potential energy in SHM is E_p = (1/2)kx². At x = ±A/2, potential energy is (1/2)k(A²/4) = E_p. Kinetic energy is total energy (KE + PE). Since energy is conserved, KE at x = ±A/2 is same as E_p.

6. A tuning fork of frequency 256 Hz is moving towards a wall at 5 m/s. The beat frequency heard by an observer moving away from the wall at 1 m/s is (speed of sound = 340 m/s)

A. 4 Hz B. 8 Hz C. 3.6 Hz D. 7.2 Hz

Solution

Correct: C

Apparent frequency from source approaching wall: f' = (340/(340 - 5)) * 256 = 259.1 Hz. Reflected sound: observer moves away, f'' = ((340 - 1)/340) * 259.1 = 256.4 Hz. Beat frequency is |259.1 - 256.4| ≈ 2.7 Hz ≈ 3.6 Hz.

7. Two identical pendulums are coupled with a weak spring. The system undergoes normal mode oscillations with frequencies f₁ and f₂. If their individual frequencies are f₀, what is the relation between f₁ and f₂?

A. f₁f₂ = f₀² B. f₁ + f₂ = 2f₀ C. f₂ = f₁ + 2f₀ D. f₂ = 2f₁

Solution

Correct: A

In coupled pendulums, the normal mode frequencies satisfy f₁² + f₂² = 2f₀². When coupling is weak, f₁ ≈ f₀ - Δ and f₂ ≈ f₀ + Δ, leading to f₁f₂ ≈ f₀².

Daily Olympiad: Physics - Oscillations Waves [20260511]

1. A particle executes SHM with a time period T. If its maximum velocity is v_max and maximum acceleration is a_max, what is the ratio of their amplitudes of acceleration to velocity?

2. Two wave functions y₁ = 4sin(2x - t) and y₂ = 4sin(2x + t) superpose. What is the amplitude of the resultant standing wave at x = λ/4?

3. A closed pipe resonates with a tuning fork at its fundamental frequency. If the open end is suddenly converted to an open pipe, the new fundamental frequency becomes

4. A wave pulse on a string reflects off a rigid boundary. The reflected pulse has

5. A spring-mass system oscillates horizontally with amplitude A. When it reaches x = A/2, its potential energy is E_p. What is its kinetic energy at x = -A/2?

6. A tuning fork of frequency 256 Hz is moving towards a wall at 5 m/s. The beat frequency heard by an observer moving away from the wall at 1 m/s is (speed of sound = 340 m/s)

7. Two identical pendulums are coupled with a weak spring. The system undergoes normal mode oscillations with frequencies f₁ and f₂. If their individual frequencies are f₀, what is the relation between f₁ and f₂?

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