Simple harmonic motion

Define simple harmonic motion.

Motion where acceleration is proportional to displacement from equilibrium and always directed towards it (a = −ω²x).

State the defining equation of SHM.

a = −ω²x

What is ω in SHM?

Angular frequency: ω = 2πf = 2π/T

Write the equation for displacement in SHM.

x = A cos(ωt)

What is the maximum speed in SHM?

vmax = ωA (occurs at equilibrium)

Write the equation for velocity at any displacement x.

v = ±ω√(A² − x²)

Where is acceleration maximum in SHM?

At maximum displacement (amplitude), a_max = ±ω²A

What is the phase difference between displacement and velocity?

π/2 radians (displacement leads velocity)

What is the phase difference between displacement and acceleration?

π radians (they are in antiphase)

Sketch/describe the x–t, v–t and a–t graphs starting from maximum positive displacement.

x–t: cosine; v–t: negative sine; a–t: negative cosine

Write the time period of a simple pendulum.

T = 2π√(l/g)

Under what condition is the simple pendulum equation valid?

Small angles only (small-angle approximation)

Does the period of a simple pendulum depend on mass or amplitude?

No (for small angles) – only on length and g

Write the time period of a mass-spring system.

T = 2π√(m/k)

How does T change if mass is quadrupled in a mass-spring system?

T doubles (T ∝ √m)

How does T change if spring constant is increased by ×4?

T halves (T ∝ 1/√k)

Where is kinetic energy maximum in SHM?

At the equilibrium position

Where is potential energy maximum in SHM?

At the amplitude (maximum displacement)

What happens to total energy in undamped SHM?

It remains constant

Define damping.

A force that reduces the amplitude of oscillations over time

Describe light damping.

Amplitude decreases gradually over many oscillations

What is critical damping?

System returns to equilibrium in the shortest time without oscillating

Why is critical damping used in car suspension?

Returns vehicle to equilibrium quickly with no bouncing

Define natural frequency.

The frequency at which a system oscillates freely with no external periodic force

Define forced oscillations.

Oscillations caused by an external periodic driving force

Define resonance.

When driving frequency equals natural frequency, producing maximum amplitude

What happens to the resonance curve as damping increases?

Peak becomes lower and broader; resonant frequency decreases slightly

What is the phase difference between driving force and displacement at resonance?

π/2 radians

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