Hooke's law, elastic strings and energy

Learning objectives

State Hooke's law as T = λx/l, where λ is the modulus of elasticity, l the natural length and x the extension, and distinguish an elastic string, which can only pull and goes slack at its natural length, from a spring, which can also push when compressed

Calculate the elastic potential energy stored in a stretched string or spring as λx²/(2l), and show that it equals the work done against the tension by integrating T with respect to the extension

Apply the work–energy principle, that the total work done by all forces equals the change in kinetic energy, and the conservation of mechanical energy where only gravity and elastic forces act, to find speeds, extensions and distances

Find the equilibrium extension e = mgl/λ of a particle hanging on an elastic string or spring, and explain why the speed of a particle moving on a taut string is greatest where its acceleration is zero

Use conservation of energy, including elastic potential energy, to find the greatest speed, the speed at a given position and the greatest extension of a particle moving vertically or horizontally on an elastic string or spring

Analyse motion in which an elastic string becomes slack, so that the particle then moves freely under gravity, by treating each phase separately and matching speed and position at the join