Revise Energy in simple harmonic motion for Cambridge 9702 (17.2)

Imagine a mass-spring system: as the mass moves, its energy continuously transforms. When it's fastest at the equilibrium position, all energy is kinetic. As it slows down moving towards maximum displacement, kinetic energy converts into potential energy, stored in the stretched or compressed spring. At the very ends of its motion, it momentarily stops, meaning all its energy is now potential, ready to pull or push it back.

$E = KE + PE$

The beauty of SHM lies in its predictable energy variations. We have specific formulas to quantify the kinetic and potential energy at any given displacement, $x$. These equations are derived from the velocity-displacement relationship in SHM and are crucial for understanding how energy distribution shifts throughout an oscillation.

Kinetic Energy (KE): $KE = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (x_0^2 - x^2)$

Potential Energy (PE): $PE = \frac{1}{2} m \omega^2 x^2$

Total Energy (E): $E = \frac{1}{2} m \omega^2 x_0^2$

The relationship between energy and displacement in SHM can be visualized with a graph. The potential energy ($PE = \frac{1}{2} m \omega^2 x^2$) is a parabola opening upwards, zero at equilibrium ($x=0$) and maximum at the amplitudes ($\pm x_0$). The kinetic energy ($KE = \frac{1}{2} m \omega^2 (x_0^2 - x^2)$) is an inverted parabola, maximum at equilibrium and zero at the amplitudes. The total energy ($E = KE + PE$) is a constant horizontal line, representing the conservation of energy. The points where the KE and PE graphs intersect are where the energy is split equally between kinetic and potential.

In the real world, no oscillation lasts forever. Damping is the process where energy is gradually lost from an oscillating system, typically converted into heat or sound. This energy dissipation causes the amplitude of the oscillations to decrease over time. The rate at which energy is removed determines the type of damping, influencing how quickly the system returns to equilibrium.

The physical cause of damping is any force that opposes motion and is non-conservative, such as air resistance or friction within the material of a spring. In a car's suspension system, shock absorbers are designed to provide critical damping, preventing the car from bouncing excessively after hitting a bump. This ensures a smooth ride and maintains tyre contact with the road.

While damping removes energy, forced oscillations involve an external force continuously supplying it. When this external driving force's frequency exactly matches the system's natural frequency, a phenomenon called resonance occurs. This leads to highly efficient energy transfer, causing a dramatic increase in the amplitude of oscillations. Think of pushing a child on a swing at just the right time!

Resonance has many important applications. In radio receivers, tuning circuits are adjusted to resonate at the frequency of a specific radio station, amplifying its signal while ignoring others. Magnetic Resonance Imaging (MRI) uses resonance of atomic nuclei in a magnetic field to create detailed images of body tissues. However, resonance can also be destructive, as famously demonstrated by the collapse of the Tacoma Narrows Bridge in 1940, where wind-induced oscillations matched the bridge's natural frequency.