Matric Physics: Simple Harmonic Motion and Waves, questions with answers

Answer: A. mg sin θ

The component of weight along the arc, mg sin θ, acts towards the mean position.

Answer: A. Restoring force

SHM requires a restoring force proportional to displacement and directed towards the mean position.

Answer: D. 10 s

T = total time / number of waves = 1800 s / 180 = 10 s.

Waveform: a sine‑shaped curve with alternate crests (highest points) and troughs (lowest points); the distance between two successive crests is the wavelength λ, and the height of a crest above the mean line is the amplitude.
In a transverse wave the particles of the medium vibrate perpendicular to the direction of wave travel. Each vibrating particle pushes/pulls its neighbour, so energy is passed from particle to particle along the medium while the particles themselves only oscillate about their mean positions — energy travels, matter does not (e.g. a wave on a rope, water ripples, light).

Diffraction (spreading of waves through a gap) is greatest when the wavelength is comparable to or larger than the width of the gap. For a given gap, waves of longer wavelength spread out more (nearly semicircular wavefronts), while waves of shorter wavelength pass through with little bending, spreading only at the edges. This is why sound (λ ≈ 1 m) bends easily round doorways but light (λ ≈ 5 × 10⁻⁷ m) travels almost straight.

T = 2π√(l/g) ⇒ l = gT²/(4π²)
On Earth (g = 10 m s⁻²): l = 10 × 2²/(4 × 9.87) = 40/39.5 ≈ 1.01 m (≈ 1 m)
On Moon (g = 1.6 m s⁻²): l = 1.6 × 4/39.5 = 6.4/39.5 ≈ 0.16 m (16 cm)
The pendulum must be about six times shorter on the Moon to keep the same period, because g there is about one‑sixth of Earth's.

Simple harmonic motion is a to‑and‑fro (vibratory) motion in which the acceleration of the body is directly proportional to its displacement from the mean position and is always directed towards the mean position.
Mathematically: a ∝ −x, i.e. a = −ω²x (or F = −kx for a mass–spring system; a = −(k/m)x). Examples: mass–spring system, simple pendulum (small angles).

A simple pendulum is a small heavy bob suspended by a light inextensible string of length l from a fixed support. When displaced through a small angle θ and released it oscillates about its mean position.
Proof of SHM: At displacement x the weight mg of the bob has two components: mg cos θ along the string (balanced by the tension) and mg sin θ along the arc, directed towards the mean position — this is the restoring force: F = −mg sin θ. For small angles sin θ ≈ θ = x/l, so
F = −(mg/l) x ⇒ ma = −(mg/l) x ⇒ a = −(g/l) x
The acceleration is proportional to the displacement and directed towards the mean position; hence the motion is simple harmonic. …

The full model answer is in the app.