Oscillation of a Spring: Restoring Force and Force Constant

A block attached to one end of a spring, with the other end fixed, and free to slide on a frictionless horizontal surface, is the simplest and most direct physical realisation of S.H.M. anywhere in this chapter.\n\nHOOKE'S LAW states that, within its elastic limit (that is, provided the spring is not stretched or compressed so far that it becomes permanently deformed), a spring displaced by a distance $x$ from its natural, unstretched length exerts a RESTORING force back towards that natural length, given by\n\n$F = -kx$\n\nHere the constant of proportionality $k$ is called the FORCE CONSTANT, or SPRING CONSTANT, of that particular spring, measured in newtons per metre (N/m). The force constant is a fixed property of a given spring, describing its stiffness: a LARGE value of $k$ means a stiff spring, one that resists being stretched or compressed strongly, requiring a large force for even a small displacement; a SMALL value of $k$ means a soft, easily-stretched spring, giving a large displacement for only a small applied force. The negative sign in Hooke's law shows that the spring force always opposes the displacement, pulling (or pushing) the block back towards the spring's natural length, whichever way the spring has actually been disturbed.\n\nSince this restoring force is directly proportional to displacement and always oppositely directed -- exactly the defining condition for S.H.M. worked out in Section 13.4 -- a block of mass $m$ attached to such a spring, once displaced from its natural length and released, executes S.H.M. with angular frequency\n\n$\omega = \sqrt{\frac{k}{m}}$\n\nand therefore with a time period\n\n$T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$\n\nA stiffer spring (larger $k$) makes the block oscillate FASTER (larger $\omega$, hence smaller $T$) for a given mass, while attaching a heavier block (larger $m$) to the very same spri …