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

## Hooke's Law and the Modulus of Elasticity[edit]

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** is *proportional* to the **tension** 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** and the **modulus of elasticity** (which is a property of the elastic string or spring independent of its length) as follows:

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 suspended vertically from one end of a light (i.e. massless) elastic string of natural length and modulus of elasticity . 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):

, which is the corresponding extension in the string. |

## Elastic Potential Energy[edit]

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 wrt the extension :

Work done to produce the extension | |