Unit 14: Simple Harmonic Motion — Long Questions

Consider a body of mass m attached to a spring of spring constant k, lying on a smooth horizontal surface, with the body at rest at the mean position O. When the body is displaced through a displacement x to a point A and released, the spring exerts an elastic restoring force on it.

By Hooke's law, the restoring force is proportional to the displacement and opposite in direction:
F = −kx

By Newton's second law of motion, F = ma. Comparing the two expressions:
ma = −kx

Since k and m are constants, this can be written as:
a = −(k/m)x, i.e. a ∝ −x

This is the defining mathematical condition of S.H.M. — the acceleration is directly proportional to the displacement and always directed towards the mean position (negative sign). Hence the motion of the mass attached to the spring is simple harmonic.

Comparing a = −(k/m)x with the standard S.H.M. equation a = −ω²x, we get:
ω² = k/m, so ω = √(k/m)

Time period T = 2π/ω = 2π√(m/k)

Frequency f = 1/T = (1/2π)√(k/m)

Consider a particle P moving with uniform angular velocity ω along the circumference of a circle of radius x₀. Let Q be the projection of P on a diameter AB of the circle, oscillating about the centre O as P revolves.

Instantaneous displacement If the radius OP sweeps out an angle θ = ωt from the initial position, then in right triangle POQ, OQ/OP = cosθ, so:
x = x₀ cos ωt ... (displacement of Q)

Instantaneous velocity The linear velocity of P is v_p = x₀ω, directed along the tangent at P. The horizontal component of this velocity equals the (horizontal) velocity of the projection Q:
v = v_p cos(90° − θ) = v_p sinθ

Using sinθ = √(1 − cos²θ) and cosθ = x/x₀:
v = x₀ω√(1 − x²/x₀²) = ω√(x₀² − x²)

At the mean position (x = 0), v = ωx₀, which is the maximum velocity.

Instantaneous acceleration The acceleration of P is the centripetal acceleration a_p = −x₀ω², directed towards the centre O. Its horizontal component gives the acceleration of Q:
a = a_p cosθ = −x₀ω²(x/x₀) = −ω²x

Since a ∝ −x, this satisfies the defining condition of S.H.M. Hence, when a particle moves uniformly around a circle, its projection on a diameter executes simple harmonic motion, with instantaneous velocity v = ω√(x₀² − x²).

A simple pendulum consists of a small heavy solid bob of mass m suspended from a rigid support by a light, inextensible string of length ℓ, free to swing in a vertical plane.

Showing the motion is S.H.M.: When the pendulum is displaced from its mean position O through a small angle θ to the extreme position P and released, a restoring force acts on it. At P, the weight mg of the bob is resolved into mg cosθ (along the string, balanced by the tension) and mg sinθ (perpendicular to the string, along the direction of motion), which provides the restoring force:
F = −mg sinθ

By Newton's second law, F = ma, so:
ma = −mg sinθ, i.e. a = −g sinθ

For small angles, sinθ ≈ θ (in radians), and θ = S/ℓ, where S is the arc length (≈ the displacement x for small θ). Therefore:
a = −(g/ℓ)x, i.e. a ∝ −x

Since acceleration is directly proportional to displacement and directed towards the mean position, the motion of a simple pendulum is simple harmonic.

Time period Comparing a = −(g/ℓ)x with a = −ω²x:
ω² = g/ℓ, so ω = √(g/ℓ)

T = 2π/ω = 2π√(ℓ/g)

Factors affecting T The time period is directly proportional to the square root of the length ℓ of the pendulum, and inversely proportional to the square root of the acceleration due to gravity g at that place. T is independent of the mass of the bob and of the amplitude (for small oscillations).

Potential energy When the spring is stretched from the mean position to a maximum displacement x₀, work is done against the restoring force, which is stored as elastic potential energy:
W = ½kx₀²

At any instantaneous displacement x, the potential energy is:
P.E. = ½kx² = ½mω²x² (using k = mω²)
P.E. is zero at x = 0 (mean position) and maximum at x = ±x₀ (extreme positions).

Kinetic energy The instantaneous velocity of a particle executing S.H.M. is v = ω√(x₀² − x²). So the kinetic energy at displacement x is:
K.E. = ½mv² = ½mω²(x₀² − x²) = ½k(x₀² − x²)
K.E. is maximum (½mω²x₀²) at the mean position (x = 0) and zero at the extreme positions (x = ±x₀).

Total energy Adding the two,
TE = K.E. + P.E. = ½k(x₀² − x²) + ½kx² = ½kx₀² = ½mω²x₀²

Since x₀, k, m and ω are all constants for a given oscillator, TE does not depend on the instantaneous displacement x (or on time); hence the total energy of the mass-spring system remains constant throughout the motion, continuously converting between kinetic and potential forms but keeping their sum fixed — this proves conservation of energy in S.H.M. The total energy is directly proportional to the square of the amplitude, x₀².

Resonance When an oscillating body is subjected to a periodic external (driving) force whose frequency equals the natural frequency of the body (or is an integral multiple of it), the amplitude of oscillation becomes very large. This phenomenon is called resonance.

Advantages (circumstances in which resonance is useful):
1. Heating/cooking food in a microwave oven — the microwaves' frequency matches the natural frequency of water molecules in food, causing them to oscillate strongly and collide with one another; the resulting friction converts kinetic energy into heat, warming the food (containers without water molecules, such as plastic or glass, do not heat up).
2. Radio tuning — turning the knob of a radio changes the natural frequency of the receiver's electrical circuit until it matches the transmission frequency of a particular station; at resonance, the energy absorbed from that station is maximum, so only that station is heard clearly.

Disadvantages (circumstances in which resonance should be avoided):
1. Bridge vibration — if the frequency of marching footsteps (or wind-induced vibration, as in the Tacoma Narrows Bridge in 1940) matches the natural frequency of a bridge, resonance produces very large amplitude vibrations that can damage or collapse the bridge.
2. Aeroplane wings — if forces such as aerodynamic forces, turbulence or engine vibration have a frequency matching the natural frequency of the wings, resonance can occur and may prove dangerous to the aircraft structure.

The quantity θ = ωt + φ, called the phase of the motion, specifies both the instantaneous displacement x = x₀cos(ωt + φ) of the oscillating body and the direction in which it is moving, at any given instant. In this way the phase describes the complete state of motion of the oscillator at a given time — not just where it is, but also which way it is heading — unlike displacement alone, which does not reveal the direction of motion.

The constant φ, called the phase constant (or phase angle), depends on the displacement and velocity of the particle at t = 0, and so it fixes the starting point of the oscillation in time.

Relation to timing and synchronization: Two oscillators having the same frequency can still differ in phase. If the phase difference between them is 0° (or 360°), they reach their extreme and mean positions at exactly the same instants — they are said to be oscillating in phase, i.e. perfectly synchronized. If the phase difference is 180°, one oscillator is at an extreme position while the other is simultaneously at the opposite extreme — they are said to be oscillating out of phase, i.e. their motions are exactly opposed in timing. Thus the phase difference between two oscillating systems tells us how their timings are related and whether their motions are synchronized or opposed.