Energy sloshing back and forth
An oscillator keeps two energy accounts. Moving, it carries kinetic energy K = \tfrac12 mv^2. Displaced, the spring stores elastic potential energy U = \tfrac12 kx^2. As the block travels, energy pours from one account into the other, but on a frictionless table the sum — the mechanical energy — never changes (conservation of mechanical energy).
The two energy stores of a spring oscillator.
Total mechanical energy is fixed by the amplitude alone.
At the turning points all the energy is potential, \tfrac12 kA^2, and the speed is zero; at equilibrium all of it is kinetic, \tfrac12 m v_{\max}^2, and x is zero. Setting those two equal, \tfrac12 m v_{\max}^2 = \tfrac12 kA^2, recovers v_{\max} = A\sqrt{k/m} = A\omega — beautifully consistent with Guide 3. This is energy in simple harmonic motion.
Speed at any displacement x, straight from energy conservation.
The energy diagram
Plot the potential energy U = \tfrac12 kx^2 against x and you get a parabola — a potential-energy diagram. The fixed total energy is a horizontal line at \tfrac12 kA^2; the gap from that line down to the parabola is the kinetic energy at each point. The block is trapped inside the bowl, rolling between the two turning points where the line meets the curve.
This bowl is the deepest reason SHM is universal. Near any stable equilibrium the potential-energy curve, whatever its true shape, looks like a parabola when you zoom in close enough. So tiny oscillations about that minimum — a rocking boat, a stretched molecular bond, a marble in a dish — are all approximately simple harmonic. The mass-spring bowl is a portrait of stable equilibrium everywhere.
The simple pendulum
A simple pendulum is a small bob on a light string of length L. Pull it aside by an angle \theta and gravity supplies the restoring pull. The component of gravity along the arc is -mg\sin\theta. For small angles, \sin\theta \approx \theta (with \theta in radians), so the restoring force becomes proportional to the displacement — and we are back to SHM.
The small-angle approximation turns the pendulum into a harmonic oscillator.
Working it through — the tangential displacement is the arc s = L\theta, and the same restoring-force logic gives a = -(g/L)\,s — the angular frequency and period fall out with g playing the role of stiffness and L the role of inertia.
Period of a simple pendulum (small-angle approximation).
Worked example and other pendulums
How long is a 'seconds pendulum'? A pendulum that ticks once each second swings one way per second, so a full cycle is T = 2.0 s. Solve T = 2\pi\sqrt{L/g} for the length, using g = 9.8 m/s².
A seconds pendulum is about one metre long — hence the height of grandfather clocks.
Beyond the ideal case: a real swinging body — a baseball bat, a leg, a hanging rod — is a physical pendulum. Its period uses its moment of inertia I and the distance d from the pivot to its centre of mass. A disc twisting back and forth on a wire is a torsional pendulum. All wear the same SHM template, just with a different 'stiffness' and 'inertia'. (The ballistic pendulum, which catches a bullet and swings up, uses these same swings to measure momentum.)
The physical pendulum: same form, now with rotational inertia I and pivot distance d.