A-level Mathematics/OCR/M3/Elastic Strings and Springs

Hooke's Law and the Modulus of Elasticity

The natural length is the length of an elastic string or spring when it is not stretched or compressed. An elastic string or spring experiences a tension when its length is greater than the natural length. In addition, a spring experiences a compression when its length is less than its natural length. To simplify our analysis, we use the term tension to refer to both types of forces (i.e. tension and compression). Also, we use the term extension to refer to the change in the length of the string or spring. Thus, the extension for a compressed spring is negative.

According to Hooke's Law, the extension $x$ is proportional to the tension $T$ applied to the elastic string or spring. Although Hooke's Law holds only up to the limit of elasticity, we may safely assume its applicability unless otherwise told. We may write this relationship in terms of the natural length $l$ and the modulus of elasticity $\lambda$ (which is a property of the elastic string or spring independent of its length) as follows:

$T={\frac {\lambda }{l}}x$

Note that the tension and the extension are in the same direction (i.e. the variables are either both positive or both negative). This should be intuitive since we are considering the force exerted on (i.e. NOT exerted by) the elastic string or spring.

If a mass attached to the end of the elastic string or spring is producing the extension, then by Newton's Third Law, the elastic string or spring exerts a force on the mass equal in magnitude and opposite in direction to its tension. To illustrate this, let us consider the system on the right.

Consider a particle P of mass $m$ suspended vertically from one end of a light (i.e. massless) elastic string of natural length $l$ and modulus of elasticity $\lambda$. The other end of the string is attached to a fixed point O. When the particle is at rest, the resultant force acting on it is zero according to Newton's Second Law. Therefore, the downward weight of the particle should balance the upward force exerted by the string on the particle (which is equal in magnitude to the tension of the string):

	$mg$	$=T$
$\Rightarrow$	$mg$	$={\frac {\lambda }{l}}x$
$\Rightarrow$	$x$	$={\frac {mgl}{\lambda }}$, which is the corresponding extension in the string.

Elastic Potential Energy

An elastic string or spring is able to store energy when it is extended (and compressed, in the case of a spring). This stored energy is termed the elastic potential energy (EPE). The EPE in an elastic string or spring is converted from the work done (by an external agent) in producing the required extension. This is just the work done against the force exerted by the elastic string or spring by virtue of its tension. Therefore, the EPE can be determined by integrating the tension $T$ wrt the extension $x$:

$\mathrm {EPE}$	$=$Work done to produce the extension $x$
	$=\int _{0}^{x}T\,dy$
	$=\int _{0}^{x}{\frac {\lambda }{l}}y\,dy$
	$={\frac {\lambda }{l}}\left[{\frac {y^{2}}{2}}\right]_{0}^{x}$

$\Rightarrow$

$\mathrm {EPE} ={\frac {\lambda }{2l}}x^{2}$