1
Study guides with 150 questions and answers on
oscillations and waves for college and university students.
Each answer includes a concise explanation, derivation, or
example-level insight.
1. What is simple harmonic motion (SHM)?
● SHM is motion where the restoring force is proportional to the negative displacement: F
= −kx. This yields x¨ + (k/m)x = 0 with solution x(t) = A cos(ωt + φ), ω = √(k/m). The
linear restoring force makes motion sinusoidal.
2. What differential equation defines SHM?
● x¨ + ω² x = 0. It arises from Newton’s second law with F = −kx. Its general solution is
sinusoidal; the frequency is set by system parameters.
3. How are period, frequency, and angular frequency related?
● ω = 2πf = 2π/T. Frequency f is cycles per second, period T is time per cycle, and ω
measures radians per second.
4. What is the total mechanical energy of a mass–spring in SHM?
● E = 1/2 kA² = 1/2 mω²A². Energy is constant: at x = ±A potential is maximal; at x = 0
kinetic is maximal.
5. Where are speed and acceleration maximal in SHM?
● Speed is maximal at x = 0; acceleration is maximal at x = ±A. Because a = −ω²x,
acceleration magnitude grows with displacement; speed is highest where potential energy
is lowest.
6. What is the phase constant in SHM?
● φ sets the initial condition. For x(0) = x0 and v(0) = v0, φ = atan2(−v0/(ωA), x0/A). It
shifts the cosine/sine curve in time.
7. How do you find amplitude from initial conditions?
● A = √(x0² + (v0/ω)²). Derived from x = A cos φ and v = −Aω sin φ at t = 0.
8. What is the velocity as a function of time in SHM?
, 2
● v(t) = −Aω sin(ωt + φ). It’s 90° out of phase with displacement.
9. What is the acceleration in SHM?
● a(t) = −ω² x(t) = −Aω² cos(ωt + φ). Acceleration is proportional to displacement with
opposite sign.
10.How does period depend on amplitude in ideal SHM?
● It does not; T = 2π/ω is amplitude-independent for linear restoring forces. Nonlinearities
introduce amplitude dependence.
11.What is the period of a mass–spring system?
● T = 2π √(m/k). Derived from ω = √(k/m).
12.What is the period of a simple pendulum (small angles)?
● T ≈ 2π √(L/g). For θ small, sin θ ≈ θ, giving SHM with ω = √(g/L).
13.What is the effective spring constant for springs in series?
● keq = (k1 k2)/(k1 + k2). Series springs extend more, lowering stiffness.
14.What about springs in parallel?
● keq = k1 + k2. Parallel springs share load, increasing stiffness.
15.What is damping?
● A dissipative force, often proportional to velocity: Fd = −bv. It removes mechanical
energy, reducing amplitude over time.
16.What equation describes damped SHM?
● x¨ + 2β x˙ + ω0² x = 0, where β = b/(2m), ω0 = √(k/m). Solution type depends on
damping ratio ζ = β/ω0.
17.What are the regimes of damping?
● Under damped (β < ω0): oscillatory with decaying amplitude; critically damped (β = ω0):
fastest non-oscillatory return; over damped (β > ω0): slow non-oscillatory return.
18.What is the under damped solution form?
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● x(t) = A e^(−βt) cos(ωd t + φ), with ωd = √(ω0² − β²). Exponential envelope multiplies
oscillation.
19.What is the quality factor Q?
● Q ≈ ω0/(2β) for weak damping. It measures sharpness of resonance and energy storage
relative to losses; higher Q means slower decay.
20.How fast does amplitude decay in a lightly damped oscillator?
● A(t) = A0 e^(−βt). The time constant τ = 1/β characterizes decay rate.
21.What is the energy decay in a damped oscillator?
● E ∝ A² ∝ e^(−2βt). Energy decays twice as fast in exponent as amplitude.
22.What is driven (forced) oscillation?
● An external periodic force F0 cos(Ωt) acts: x¨ + 2β x˙ + ω0² x = (F0/m) cos(Ωt).
Steady-state response oscillates at Ω.
23.What is mechanical impedance?
● Z(Ω) = m(ω0² − Ω² + 2iβΩ)/iΩ for displacement-velocity form; often use complex
amplitude methods. It relates force to velocity or displacement in frequency domain.
24.What is resonance?
● Maximal steady-state amplitude occurs near Ω ≈ ωr = √(ω0² − 2β²) for light damping.
Energy input from the drive matches loss.
25.What is the amplitude response of a driven damped oscillator?
● X(Ω) = (F0/m)/√((ω0² − Ω²)² + (2βΩ)²). Shows resonance peak that narrows as β
decreases.
26.What is the phase of the driven response?
● tan δ = (2βΩ)/(ω0² − Ω²). Phase shifts from 0 to π across resonance, passing π/2 at Ω =
ω0 for light damping.
27.What is the power absorbed at frequency Ω?
● Pavg = (1/2) F0 V0 cos φ, where φ is phase between force and velocity. Peaks at
resonance for small damping.
Study guides with 150 questions and answers on
oscillations and waves for college and university students.
Each answer includes a concise explanation, derivation, or
example-level insight.
1. What is simple harmonic motion (SHM)?
● SHM is motion where the restoring force is proportional to the negative displacement: F
= −kx. This yields x¨ + (k/m)x = 0 with solution x(t) = A cos(ωt + φ), ω = √(k/m). The
linear restoring force makes motion sinusoidal.
2. What differential equation defines SHM?
● x¨ + ω² x = 0. It arises from Newton’s second law with F = −kx. Its general solution is
sinusoidal; the frequency is set by system parameters.
3. How are period, frequency, and angular frequency related?
● ω = 2πf = 2π/T. Frequency f is cycles per second, period T is time per cycle, and ω
measures radians per second.
4. What is the total mechanical energy of a mass–spring in SHM?
● E = 1/2 kA² = 1/2 mω²A². Energy is constant: at x = ±A potential is maximal; at x = 0
kinetic is maximal.
5. Where are speed and acceleration maximal in SHM?
● Speed is maximal at x = 0; acceleration is maximal at x = ±A. Because a = −ω²x,
acceleration magnitude grows with displacement; speed is highest where potential energy
is lowest.
6. What is the phase constant in SHM?
● φ sets the initial condition. For x(0) = x0 and v(0) = v0, φ = atan2(−v0/(ωA), x0/A). It
shifts the cosine/sine curve in time.
7. How do you find amplitude from initial conditions?
● A = √(x0² + (v0/ω)²). Derived from x = A cos φ and v = −Aω sin φ at t = 0.
8. What is the velocity as a function of time in SHM?
, 2
● v(t) = −Aω sin(ωt + φ). It’s 90° out of phase with displacement.
9. What is the acceleration in SHM?
● a(t) = −ω² x(t) = −Aω² cos(ωt + φ). Acceleration is proportional to displacement with
opposite sign.
10.How does period depend on amplitude in ideal SHM?
● It does not; T = 2π/ω is amplitude-independent for linear restoring forces. Nonlinearities
introduce amplitude dependence.
11.What is the period of a mass–spring system?
● T = 2π √(m/k). Derived from ω = √(k/m).
12.What is the period of a simple pendulum (small angles)?
● T ≈ 2π √(L/g). For θ small, sin θ ≈ θ, giving SHM with ω = √(g/L).
13.What is the effective spring constant for springs in series?
● keq = (k1 k2)/(k1 + k2). Series springs extend more, lowering stiffness.
14.What about springs in parallel?
● keq = k1 + k2. Parallel springs share load, increasing stiffness.
15.What is damping?
● A dissipative force, often proportional to velocity: Fd = −bv. It removes mechanical
energy, reducing amplitude over time.
16.What equation describes damped SHM?
● x¨ + 2β x˙ + ω0² x = 0, where β = b/(2m), ω0 = √(k/m). Solution type depends on
damping ratio ζ = β/ω0.
17.What are the regimes of damping?
● Under damped (β < ω0): oscillatory with decaying amplitude; critically damped (β = ω0):
fastest non-oscillatory return; over damped (β > ω0): slow non-oscillatory return.
18.What is the under damped solution form?
, 3
● x(t) = A e^(−βt) cos(ωd t + φ), with ωd = √(ω0² − β²). Exponential envelope multiplies
oscillation.
19.What is the quality factor Q?
● Q ≈ ω0/(2β) for weak damping. It measures sharpness of resonance and energy storage
relative to losses; higher Q means slower decay.
20.How fast does amplitude decay in a lightly damped oscillator?
● A(t) = A0 e^(−βt). The time constant τ = 1/β characterizes decay rate.
21.What is the energy decay in a damped oscillator?
● E ∝ A² ∝ e^(−2βt). Energy decays twice as fast in exponent as amplitude.
22.What is driven (forced) oscillation?
● An external periodic force F0 cos(Ωt) acts: x¨ + 2β x˙ + ω0² x = (F0/m) cos(Ωt).
Steady-state response oscillates at Ω.
23.What is mechanical impedance?
● Z(Ω) = m(ω0² − Ω² + 2iβΩ)/iΩ for displacement-velocity form; often use complex
amplitude methods. It relates force to velocity or displacement in frequency domain.
24.What is resonance?
● Maximal steady-state amplitude occurs near Ω ≈ ωr = √(ω0² − 2β²) for light damping.
Energy input from the drive matches loss.
25.What is the amplitude response of a driven damped oscillator?
● X(Ω) = (F0/m)/√((ω0² − Ω²)² + (2βΩ)²). Shows resonance peak that narrows as β
decreases.
26.What is the phase of the driven response?
● tan δ = (2βΩ)/(ω0² − Ω²). Phase shifts from 0 to π across resonance, passing π/2 at Ω =
ω0 for light damping.
27.What is the power absorbed at frequency Ω?
● Pavg = (1/2) F0 V0 cos φ, where φ is phase between force and velocity. Peaks at
resonance for small damping.
