Hooke's Law (College Board AP® Physics 1: Algebra-Based): Study Guide

Ideal spring

• When a spring is neither stretched nor compressed, it is said to be in equilibrium

  • The net force on the spring is zero

  • Its length at its equilibrium position is known as the relaxed length

• When a spring is stretched or compressed, its change in length (extension or compression), $\mathrm{\Delta }\overrightarrow{x}$ , is the displacement from its equilibrium position

  • The spring will exert a force, ${\overrightarrow{F}}_s$ , towards its equilibrium position

Force exerted by a spring

• An ideal spring can be defined as:

A spring that has negligible mass and exerts a force that is proportional to the change in its length as measured from its relaxed length

• This can be expressed as:

${\overrightarrow{F}}_s\propto -\mathrm{\Delta }\overrightarrow{x}$

• Where:

  • ${\overrightarrow{F}}_s$ = force exerted by a spring, in $\mathrm{N}$

  • $\mathrm{\Delta }\overrightarrow{x}$ = change in length of the spring, in $\mathrm{m}$

• The minus sign shows that the force exerted by a spring is a restoring force as it acts toward the equilibrium position

Hooke's law

• When an object is placed on an ideal spring, the spring will exert a force on the object

• For an ideal spring:

${\overrightarrow{F}}_s\propto -\mathrm{\Delta }\overrightarrow{x}$

• Where:

  • ${\overrightarrow{F}}_s$ = force exerted by a spring, in $\mathrm{N}$

  • $\mathrm{\Delta }\overrightarrow{x}$ = change in length of the spring, in $\mathrm{m}$

• This can be re-written using a proportionality constant, called the spring constant, to show the magnitude of this force

• This equation is known as Hooke's law:

${\overrightarrow{F}}_s=-k\mathrm{\Delta }\overrightarrow{x}$

• Where:

  • ${\overrightarrow{F}}_s$ = force exerted by a spring, in $\mathrm{N}$

  • $k$ = spring constant, in $\mathrm{N}/\mathrm{m}$

  • $\mathrm{\Delta }\overrightarrow{x}$ = change in length, in $\mathrm{m}$

• The force that is exerted by an ideal spring on an object is a restoring force

• Hooke's law applies to both extensions and compressions:

  • The extension of an object is determined by how much it has increased in length

  • The compression of an object is determined by how much it has decreased in length

• The extension, $\mathrm{\Delta }x$, is the difference between the relaxed and stretched length 

$\mathrm{extension}=\mathrm{stretched}\mathrm{length}-\mathrm{relaxed}\mathrm{length}$

Force exerted on an object by an ideal spring

Force-versus-extension graphs

• The relationship between force and the change in length from its relaxed length is shown on a force-versus-extension graph

  • For an ideal spring, this is a linear relationship shown as a straight-line graph which passes through the origin

Calculating the spring constant from a force-versus-extension graph

• The stiffer the spring, the greater the spring constant and vice versa

  • This means that more force is required per meter of extension compared to a less stiff spring

• When force is plotted against extension, the steeper the slope of the line, the larger the spring constant and the stiffer the spring

Combination of springs

• Springs can be combined in different ways:

  • In series (end-to-end)

  • In parallel (side-by-side)

• The effective spring constant, $k_{eff}$ , is a measure of how much force the combination of springs exerts per unit of extension or compression

Springs in parallel

• For springs in parallel, the extension from the equilibrium position, $\mathrm{\Delta }\overrightarrow{x}$, is the same for both springs

• The net force exerted by the two springs, ${\overrightarrow{F}}_{eff}$ , is:

${\overrightarrow{F}}_{eff}={\overrightarrow{F}}_1+{\overrightarrow{F}}_2$

• Using Hooke's law, the force exerted by spring 1 is:

${\overrightarrow{F}}_1=-k_1\mathrm{\Delta }\overrightarrow{x}$

• The force exerted by spring 2 is:

${\overrightarrow{F}}_2=-k_2\mathrm{\Delta }\overrightarrow{x}$

• Substituting:

${\overrightarrow{F}}_{eff}=\left(-k_1\mathrm{\Delta }\overrightarrow{x}\right)+\left(-k_2\mathrm{\Delta }\overrightarrow{x}\right)=-\left(k_1+k_2\right)\mathrm{\Delta }\overrightarrow{x}$

• Therefore, the effective spring constant for springs in parallel, $k_{eff}$, is:

  $k_{eff}=k_1+k_2$

Springs in series

• For springs in series, the total extension, $\mathrm{\Delta }\overrightarrow{x}$ is:

$\mathrm{\Delta }\overrightarrow{x}=\mathrm{\Delta }{\overrightarrow{x}}_1+\mathrm{\Delta }{\overrightarrow{x}}_2$

• Where:

  • $\mathrm{\Delta }{\overrightarrow{x}}_1$ is the extension of spring 1, measured in $\mathrm{m}$

  • $\mathrm{\Delta }{\overrightarrow{x}}_2$ is the extension of spring 2, measured in $\mathrm{m}$

• The force is evenly distributed, so the force, $F$, exerted by each spring is equal

• The extensions can be written in terms of force and spring constants using Hooke's law:

$\mathrm{\Delta }\overrightarrow{x_1}=-\frac{{\displaystyle \overrightarrow{F}}}{k_1}$

$\mathrm{\Delta }\overrightarrow{x_2}=-\frac{{\displaystyle \overrightarrow{F}}}{k_2}$

• Therefore

$\mathrm{\Delta }\overrightarrow{x}=\left(-\frac{{\displaystyle \overrightarrow{F}}}{k_1}\right)+\left(-\frac{{\displaystyle \overrightarrow{F}}}{k_2}\right)=-\left(\frac{1}{k_1}+\frac{1}{k_2}\right)\overrightarrow{F}$

• Rearranging this in the format of Hooke's law for a single spring:

$\overrightarrow{F}=-k\mathrm{\Delta }\overrightarrow{x}$

$\overrightarrow{F}=-\left(\frac{1}{\frac{1}{k_1}+\frac{1}{k_2}}\right)\mathrm{\Delta }\overrightarrow{x}=-\left(\frac{k_1{k}_2}{k_1+k_2}\right)\mathrm{\Delta }\overrightarrow{x}$

• Therefore, the effective spring constant, $k_{eff}$, is:

$k_{eff}=\frac{k_1{k}_2}{k_1+k_2}$