15.1-15.5 - Elasticity, spring forces and energy

Forces change shape

Stretching, compressing or bending changes an object's shape. This is called distortion or deformation. More than one force is needed because forces must act on different parts of the object to pull them apart, push them together or turn them relative to one another.

• To stretch a hanging spring, the support pulls upwards on its top while the load pulls downwards on its bottom.
• To compress a spring, forces act inwards at opposite ends.
• To bend a ruler supported at both ends, the supports push upwards while another force pushes downwards between them.

Once a deformed object has settled and is stationary, these forces may balance overall. They still act at different places and create tension, compression or bending inside the object. One unopposed force would mainly change the object's motion rather than hold it in a steadily deformed shape.

A spring can be elastic over a safe range and become inelastically distorted if it is overloaded. Elastic does not mean the same as linear. An elastic band may have a curved, non-linear force-extension relationship and still return to its original length when unloaded. Recovery after unloading tests elasticity; graph shape tests linearity.

Extension and spring constant

The length shown on a ruler is not automatically the extension. Measure the unloaded length first, then subtract it from the loaded length:

The symbol $x$ represents extension, measured in metres (m). A spring that is 0.180 m long before loading and 0.260 m long after loading has an extension of $0.260-0.180=0.080\text{ m}$.

Within the linear elastic region, force and extension are related by:

The spring constant describes stiffness. A larger value of $k$ means that more force is required for each metre of extension, so the spring is stiffer.

Worked example. A spring has an unloaded length of 18.0 cm. A force of 3.2 N increases its length to 26.0 cm. Calculate its spring constant.

First find the extension and convert it to metres:

Rearrange $F=kx$:

The answer is given to two significant figures, matching the force and extension data. A force of 3.2 N producing an 8.0 cm extension is consistent with a relatively flexible laboratory spring.