Simple harmonic systems (A-level only) Flashcards – AQA A-level Physics (7408)

What is a simple pendulum?

A suspended mass oscillating about its lowest equilibrium position.

What provides a simple pendulum’s restoring acceleration, and how does it depend on angular displacement $\theta$?

The component of gravity along its path provides an acceleration proportional to $-\sin\theta$.

Why is a pendulum’s motion approximately simple harmonic at small angles?

For small angles in radians, $\sin\theta\approx\theta$, so the restoring acceleration is approximately proportional to displacement.

What is the period of a simple pendulum in the small-angle approximation?

$T=2\pi\sqrt{\frac{l}{g}}$, where $l$ is the pendulum length and $g$ is the gravitational field strength.

How does increasing a simple pendulum’s length affect its period?

A longer pendulum has a longer period.

Which two quantities do not affect a simple pendulum’s period in the small-angle approximation?

The mass and the amplitude.

What acceleration–displacement relationship defines simple harmonic motion?

$a=-\omega^2x$: acceleration is proportional to displacement from equilibrium and directed towards equilibrium.

How can the period of an oscillator be derived from its restoring acceleration?

Obtain its restoring acceleration, identify $\omega^2$ as the positive coefficient of displacement, then use $T=\frac{2\pi}{\omega}$.

What is the small-angle approximation for sine, and under what condition is it valid?

$\sin\theta\approx\theta$ when $\theta$ is small and measured in radians.

Why can the small-angle approximation help establish approximate simple harmonic motion?

It can yield a restoring acceleration proportional to displacement.

How does mechanical energy change in undamped simple harmonic motion?

Energy transfers between kinetic and potential energy, while total mechanical energy remains constant.

What are the speed, kinetic energy and potential energy at equilibrium in simple harmonic motion?

Speed and kinetic energy are maximum; potential energy is minimum.

What are the speed, kinetic energy and potential energy at either extreme displacement in simple harmonic motion?

Speed and kinetic energy are zero; potential energy is maximum.

What is the potential energy at displacement $x$ in simple harmonic motion, taking potential energy as zero at equilibrium?

$E_p = \frac{1}{2}m\omega^2x^2$

What is the kinetic energy at displacement $x$ in simple harmonic motion of amplitude $A$, taking potential energy as zero at equilibrium?

$E_k = \frac{1}{2}m\omega^2(A^2-x^2)$

What is the total mechanical energy of simple harmonic motion of amplitude $A$, taking potential energy as zero at equilibrium?

$E = \frac{1}{2}m\omega^2A^2$

If $x=A\cos(\omega t)$ in simple harmonic motion, how do potential and kinetic energy vary with time?

$E_p \propto \cos^2(\omega t)$ and $E_k \propto \sin^2(\omega t)$.

How many cycles does each of the kinetic and potential energies complete during one oscillation of simple harmonic motion?

Two cycles each.

What is damping in an oscillator?

The dissipation of the oscillator’s mechanical energy to its surroundings, causing its amplitude to decrease.

How does a lightly damped system behave?

It continues to oscillate with a gradually decreasing amplitude.

How does a heavily damped system return towards equilibrium?

It returns without oscillating.

What boundary does critical damping represent?

The boundary between oscillatory and non-oscillatory return to equilibrium.

What is special about the return to equilibrium under critical damping?

It is the fastest return without oscillation.