Oscillations · Lesson 8 of 8
Chapter Summary and Practice
“Connect the chapter’s motion, force and energy descriptions, review formula conditions, and solve mixed problems.”
• Choose correctly between periodicity, the SHM displacement law and the restoring-force criterion. • Connect amplitude and initial conditions to velocity, acceleration and energy. • Select appropriate spring, pendulum and fluid period formulas with their assumptions. • Use units and limiting cases to test an answer. • Solve mixed problems and identify common phase, sign, period and coordinate mistakes.
One motion, several connected descriptions
Begin with the physical system and choose its equilibrium coordinate. For an ideal linear oscillator, a restoring force F = −kx implies acceleration a = −(k/m)x and a sinusoidal displacement. Position and initial velocity determine amplitude and phase. Differentiation gives the signed velocity and acceleration; conservation gives the same speed magnitude without finding time. These are complementary descriptions of one motion. A repeating graph is a weaker statement: nonsinusoidal periodic motion can contain several harmonics and need not satisfy one constant linear force law.
The spring, pendulum and fluid examples illustrate the same method with different force mechanisms and inertias. A small-angle pendulum is approximately harmonic in its angle or arc displacement. A floating cylinder’s buoyancy change and a uniform U-tube’s pressure imbalance can give linear restoring forces in their ideal models. The conditions are part of each formula, rather than optional caveats to attach after calculating.
Formula reference with conditions
| Relation | Meaning and use | Condition |
|---|---|---|
| f = 1/T; ω = 2πf | f in Hz; ω in rad s⁻¹; T in s | T is the least full repeat interval |
| x = A cos(ωt + φ) | Displacement about equilibrium | Constant A and ω; signed equilibrium coordinate |
| v = −ωA sin(ωt + φ) | Signed velocity | Differentiate with radian argument |
| a = −ω²x; F = −mω²x | Restoring acceleration and force | Constant ω, constant mass |
| |v| = ω√(A² − x²) | Speed without finding time | |x| ≤ A; direction determined separately |
| vmax = ωA; |a|max = ω²A | Kinematic maxima | At equilibrium and extremes, respectively |
| U = kx²/2; K = k(A² − x²)/2 | Potential and kinetic energies | Conservative linear force; U(0) = 0 |
| E = kA²/2 = mω²A²/2 | Total mechanical energy | No damping or unaccounted external work |
| T = 2π√(m/k_eff) | Linear oscillator period | Correct effective stiffness and moving mass |
| k_eff = k₁ + k₂ | Mass between opposed fixed springs | Each spring length changes by the mass displacement |
| T = 2π√(m/(2k)) | Two equal free masses on one spring | m is either mass; centre of mass initially at rest |
| T ≈ 2π√(L/g) | Simple pendulum period | Point bob, massless string, small radians |
| geff = √(g² + V⁴/R²) | Local turning-car effective gravity | Perpendicular accelerations; locally uniform approximation |
| T = 2π√(ρh/(ρl g)) | Floating-cylinder period | Partial immersion; fluid inertia and damping neglected |
| T = 2π√(ℓ/(2g)) | Uniform U-tube period | ℓ is full moving liquid length; equal open-end pressures |
| Quantity | SI unit | Dimensions |
|---|---|---|
| Displacement x and amplitude A | m | Length |
| Period T | s | Time |
| Frequency f | Hz = s⁻¹ | Time⁻¹ |
| Angular frequency ω | rad s⁻¹ | Time⁻¹ (radian is dimensionless) |
| Phase and phase constant | rad | Dimensionless |
| Stiffness k | N m⁻¹ = kg s⁻² | Mass × Time⁻² |
| Energy K, U, E | J = kg m² s⁻² | Mass × Length² × Time⁻² |
Comparisons that prevent common errors
| Possible confusion | Correct distinction |
|---|---|
| Period versus centre-crossing interval | A centre is crossed twice per full oscillation; direction must repeat too |
| Amplitude versus extreme separation | Extreme separation or piston stroke is 2A |
| Velocity versus speed | Velocity is signed; speed is nonnegative |
| Acceleration versus velocity | At an extreme v = 0 but |a| is maximum; at centre a = 0 but |v| is maximum |
| Phase from position only | The direction of velocity normally selects between two possible phases |
| Energy period versus displacement period | K and U repeat after T/2; x repeats after T |
| Harmonic sum versus one SHM | Equal-frequency terms combine into one sinusoid; different harmonics generally do not |
| Exact periodicity of a frequency sum | A common period requires commensurate frequencies, not an integer-only frequency ratio |
| Gravity-shifted equilibrium | Use displacement from equilibrium, not full vertical spring extension |
| Pendulum frequency versus angular speed along its arc | The natural frequency is nearly constant; the bob’s instantaneous angular speed changes |
| Any restoring force versus a linear restoring force | SHM needs proportionality to displacement, not only a return tendency |
| Ideal versus damped oscillations | A decaying envelope is not exact constant-amplitude SHM |
For fixed m and k, changing the energy changes amplitude but leaves the ideal spring period unchanged. Likewise, the small-angle pendulum period is approximately independent of amplitude and bob mass, but it depends on length and gravity. This differs from gravitational orbital periods, which depend on orbit size through Kepler’s third law. Uniform circular motion is a useful reference for projection; the actual force on a linear oscillator stays along a line and varies in magnitude. In a two-dimensional isotropic harmonic field, equal perpendicular amplitudes with a quarter-cycle phase offset can instead produce an actual circular orbit.
Worked synthesis before independent assessment
Problem
A 0.50 kg mass is on a 200 N m⁻¹ spring. At time zero x₀ = 0.030 m and v₀ = −0.80 m s⁻¹. Find ω, A, a quadrant-correct time equation, total energy and speed at x = 0.025 m.
- 1.ω = √(200/0.50) = 20 rad s⁻¹. Since v₀/ω = −0.040 m, A = √(0.030² + 0.040²) = 0.050 m.
- 2.cos φ = 0.030/0.050 = 0.6 and sin φ = −v₀/(Aω) = 0.8. Thus φ = atan2(0.8,0.6) = 0.927295 rad, in the first quadrant.
- 3.The time equation is x = 0.050 cos(20t + 0.927295) m. Its initial velocity is −(20)(0.050)(0.8) = −0.80 m s⁻¹, checking the quadrant.
- 4.E = kA²/2 = 0.250 J. Independently at time zero, U₀ = 200 × 0.030²/2 = 0.090 J and K₀ = 0.50 × 0.80²/2 = 0.160 J, again summing to 0.250 J.
- 5.At x = 0.025 m, U = 0.0625 J, K = 0.1875 J and speed = √(2K/m) = √0.75 = 0.866025 m s⁻¹. The first crossing is leftward, so its velocity there is negative.
- 6.The period is 2π/20 ≈ 0.314159 s regardless of the phase or the energy for this fixed linear system.
Problem
A 1.0 m small-angle pendulum and a uniform U-tube with total liquid length 1.0 m operate under g = 9.8 m s⁻². Compare their periods and predict what happens if each relevant length is quadrupled.
- 1.The pendulum period is TP = 2π√(1/9.8) = 2.00709 s.
- 2.The U-tube period is TU = 2π√[1/(2 × 9.8)] = 1.41923 s. Thus TU = TP/√2, because its restoring coefficient contains the two moving surface contributions.
- 3.Quadrupling L or ℓ doubles the corresponding period, since each period is proportional to the square root of its own length.
- 4.Reducing g would lengthen both periods. Changing bob mass or ideal U-tube liquid density would not affect their periods under these models.
- 5.The two length symbols have different meanings: pendulum L is pivot to bob centre, whereas U-tube ℓ includes the whole liquid path through the bend.
Before trusting any result, check dimensions, sign and limits. A spring period must scale as the square root of inertia divided by stiffness. A force at negative x must be positive in the restoring model. A displaced pendulum with very small amplitude must recover the small-angle limit. A fluid formula must use the chosen level coordinate consistently, including the factor of two between U-tube surface difference and one surface displacement. These checks often reveal an error before a final number is calculated.
Quick Quiz
Quiz
An oscillator has x > 0 and v < 0. Which combination is consistent with ideal SHM?
It is moving toward equilibrium, with acceleration in the same negative direction, so speed and K increase.
Which changes the ideal spring period for fixed mass?
The period depends on m/k; phase, time origin and amplitude or energy do not change it in the ideal model.
Which statement about a uniform U-tube is correct?
The connected incompressible column moves; m = ρl Aℓ.
A graph is periodic with a triangular waveform. What follows?
Periodic does not mean sinusoidal or governed by one linear restoring law.
For K = U in a conservative oscillator, what is |x|/A?
U/E = x²/A² = 1/2.
Practice Problems
- A 0.40 kg spring oscillator has k = 160 N m⁻¹, x₀ = −0.03 m and v₀ = +0.80 m s⁻¹. Determine ω, T, A, the cosine phase quadrant, x(t), E and a₀. Verify E using both initial kinetic and potential energy.
- For x = 4 cos(πt + π/3) cm, find f, vmax and |a|max with units. Determine the speed when x = 2 cm, giving both possible velocity signs and the corresponding energy fractions.
- Compare the periods of a mass m on one spring k, the same mass between two opposed fixed springs k, and two equal free masses m joined by one spring k. Derive the effective restoring equation rather than matching formulas by appearance.
- Find the length of a pendulum with a 2 s full period on Earth. Predict its period on the Moon using gEarth = 9.8 m s⁻² and gMoon = 1.7 m s⁻². State the small-angle assumption and explain the tick convention.
- Derive periods for a cylinder of height 0.10 m and density 250 kg m⁻³ floating in water of density 1000 kg m⁻³, and for a uniform U-tube with full liquid length 0.80 m. Use g = 9.8 m s⁻² and explain why their density dependences differ.
- For a car of speed V on a radius-R track, derive the locally approximated pendulum period about its outward-tilted equilibrium using vector effective gravity. State why scalar subtraction of accelerations would be incorrect.
- Classify sin(ωt) − cos(ωt), sin²(ωt), sin³(ωt), sin(2ωt) + cos(3ωt), and exp(−ωt). Give centres and fundamental periods where appropriate and distinguish periodicity from SHM.
- Draw displacement, velocity, acceleration, kinetic-energy and potential-energy graphs for an ideal oscillator released from +A. Mark the first period and identify the different repeat intervals. Explain why a decaying real oscillation is not an exact example of these graphs.
Key Takeaways
• Identify equilibrium and the displacement coordinate before selecting any formula. • One ideal oscillator connects a sinusoidal displacement, a linear restoring force and a constant mechanical energy. • Signed initial conditions determine phase; energy provides speed magnitude and amplitude. • Spring, pendulum and fluid periods share a restoring-force method but use different inertias and assumptions. • Units, signs, geometry and limiting cases are part of a correct solution.
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Oscillations in Fluids
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