Simple harmonic motion and resonance

The conditions and equations for simple harmonic motion, oscillation graphs, energy, damping and resonance.

Understand that the condition for simple harmonic motion is $F = -kx$ , and hence understand how to identify situations in which simple harmonic motion will occur

Be able to use the equations $a = -\omega^2 x$ , , , , and as applied to a simple harmonic oscillator

Be able to use the equations for a simple harmonic oscillator $T = 2\pi\sqrt{\frac{m}{k}}$ and a simple pendulum $T = 2\pi\sqrt{\frac{l}{g}}$

Be able to draw and interpret a displacement-time graph for an oscillating object and know that the gradient at a point gives the velocity at that point

Be able to draw and interpret a velocity-time graph for an oscillating object and know that the gradient at a point gives the acceleration at that point

Understand what is meant by resonance

CORE PRACTICAL 16: Determine the value of an unknown mass using the resonant frequencies of the oscillation of known masses

Understand how to apply conservation of energy to damped and undamped oscillating systems

Understand the distinction between free and forced oscillations

Understand how the amplitude of a forced oscillation changes at and around the natural frequency of a system and know, qualitatively, how damping affects resonance

Understand how damping and the plastic deformation of ductile materials reduce the amplitude of oscillation

A mass of $0.40\,\text{kg}$ hangs from a spring of stiffness $25\,\text{N m}^{-1}$ . It is pulled down $5.0\,\text{cm}$ from equilibrium and released, so it oscillates with simple harmonic motion.

(a) Calculate the period of oscillation.

(b) Calculate the maximum speed of the mass.

$T = 2\pi\sqrt{m/k} = 2\pi\sqrt{0.40/25}$ (1)

Tall buildings can sway when the wind or an earthquake makes them oscillate. Engineers fit large dampers to reduce this.

Explain what is meant by resonance, why it can be dangerous for a building, and how damping reduces the danger.

Indicative content:

Resonance occurs when a system is forced to oscillate at (or near) its natural frequency (1)
Energy is transferred to the system most efficiently, so the amplitude becomes very large (1)
A building has a natural frequency; if the driving frequency (e.g. from earthquake waves or wind) matches it, it resonates (1)
Large amplitude oscillations produce large stresses that can damage or collapse the structure (1)
Damping removes energy from the oscillating system (e.g. by doing work against friction/viscous forces, transferring it to thermal energy) (1)
Damping reduces the maximum amplitude at resonance and broadens the resonance peak (and lowers the resonant frequency slightly) (1)

Award 1 mark per relevant point, max 6.

Specification points

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Sample questions