Describe simple examples of free oscillations such as the simple pendulum, spring-mass system and torsional pendulum.

The displacement $x$ of a simple oscillator as a function of time $t$ is given by:

$$x = x_0 \sin(\omega t)$$

where $x_0$ is the amplitude and $\omega$ is the angular frequency.

Velocity $v$ is the first derivative of displacement with respect to time:

$$v = \frac{dx}{dt}$$ $$v = \frac{d}{dt} (x_0 \sin(\omega t))$$

Using the chain rule, $\frac{d}{dt} (\sin(at)) = a \cos(at)$:

$$v = x_0 \omega \cos(\omega t)$$

Acceleration $a$ is the first derivative of velocity with respect to time (or the second derivative of displacement):

$$a = \frac{dv}{dt}$$ $$a = \frac{d}{dt} (x_0 \omega \cos(\omega t))$$

Using the chain rule, $\frac{d}{dt} (\cos(at)) = -a \sin(at)$:

$$a = x_0 \omega (-\omega \sin(\omega t))$$ $$a = -x_0 \omega^2 \sin(\omega t)$$

From Step 1, we know that $x = x_0 \sin(\omega t)$. We can substitute this into the expression for $a$ from Step 3:

$$a = -\omega^2 (x_0 \sin(\omega t))$$ $$a = -\omega^2 x$$

This proves that for simple oscillations, the acceleration is directly proportional to the negative of the displacement, which is the defining characteristic of simple harmonic motion.