Wave behaviour

1. The graph gives F = −20x, so a = F/m = −40x: acceleration is proportional to displacement and directed toward equilibrium.
2. Comparing a = −ω²x gives ω = √40 rad s⁻¹ and T = 2π/ω ≈0.993 s.
3. The amplitude is 0.10 m. From energy conservation or vmax = ωA, the maximum speed is 0.632 m s⁻¹ at equilibrium.
4. At x = 0.060 m, acceleration is −40(0.060) = −2.4 m s⁻², toward equilibrium.
5. Energy gives v² = ω²(A²−x²) = 40(0.0100−0.0036) = 0.256; speed is √0.256 ≈0.506 m s⁻¹, with direction depending on the motion phase.
6. A driving frequency near the natural frequency can produce a large response. Damping transfers mechanical energy to the surroundings and reduces and broadens the resonance peak, useful when large vibrations would cause damage.

Answer (a) ω = √40 ≈6.32 rad s⁻¹; T ≈0.993 s; vmax ≈0.632 m s⁻¹. (b) Speed magnitude ≈0.506 m s⁻¹; acceleration = −2.4 m s⁻².

Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.

1. One complete spatial cycle spans 0.80 m, and maximum displacement from equilibrium is 3.0 cm.
2. Frequency is cycles/time = 5/2.0 = 2.5 Hz; T = 1/f = 0.40 s; propagation speed is fλ = 2.0 m s⁻¹.
3. A single spatial profile can move either left or right. A later snapshot or local displacement–time information is needed to establish direction.
4. Superposition adds signed displacements: +3.0+(−2.0) = +1.0 cm upward.
5. Partial cancellation describes displacement at that location and instant. It does not mean the waves permanently destroy each other.
6. The pulses continue in the ideal linear model. Energy is redistributed between kinetic and elastic forms; displacement alone is insufficient to determine all local energy during overlap.

Answer (a) A = 3.0 cm; λ = 0.80 m; T = 0.40 s; f = 2.5 Hz; v = 2.0 m s⁻¹. (b) Resultant displacement is +1.0 cm; in the ideal linear medium both pulses continue through each other.

Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.

1. Snell's law gives nair sin i = nmaterial sin r. The graph gradient sin i/sin r is 1.50.
2. Wave speed in the material is c/n = 2.0×10⁸ m s⁻¹. The critical angle obeys sin θc = 1/1.50, giving about 41.8°.
3. The internal angle 45° exceeds θc and light travels from higher to lower index, meeting both conditions for total internal reflection.
4. The boundary must transmit the same oscillation frequency. Since v = fλ and speed drops by a factor of 1.50, wavelength also drops by that factor.
5. A ray arriving below the critical angle can refract into the surrounding lower-index material, so a transparent tube does not trap all entry directions.
6. Real fibres have cladding, losses and bending limits. A tight bend can change incidence angles and allow previously guided light to escape.

Answer (a) n = 1.50; speed = 2.0×10⁸ m s⁻¹; critical angle ≈41.8°; total internal reflection occurs at 45°. (b) Frequency stays fixed; speed and wavelength fall on entry; only rays meeting the total-internal-reflection condition are trapped in the ideal model.

Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.

1. For two-slit interference, w = λD/d. The slope w/D is 2.0 mm m⁻¹ = 0.0020.
2. λ = slope×d = 0.0020(0.00030) = 6.0×10⁻⁷ m.
3. Doubling d halves w at fixed λ and D. At 1.5 m, the original 3.0 mm spacing becomes 1.5 mm.
4. First-to-eleventh fringe centres contain ten intervals, not eleven. A 30 mm span is less sensitive to a fixed endpoint error than one 3 mm interval.
5. Keep wavelength and slit separation fixed while changing D; proportional spacing supports the wave model. Then change d while keeping D fixed and test the inverse relationship.
6. Record repeat endpoint readings, keep the screen perpendicular to the central ray and shield stray light. A ruler artefact would not track both controlled physical changes.

Answer (a) Wavelength = 600 nm; doubled-separation fringe spacing = 1.5 mm. (b) Measure a long span over many intervals and vary a physical input with a quantitative prediction.

Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.