spring force

Contact force between to objects connected via a spring, or between object and a fixed point, considered in butiran.

In the field of technology, spring can be considered as elastic machine component that is able to deflect under load in a prescribed manner and to recover its initial shape when unloaded, which includes several spring types such as helical spring, leaf spring, torsion-bar spring, air spring, and hydraulic spring ¹. Restoring force that tends to restore spring to its original shape is known as spring force, which is a contact force that can also be found in elastic materials ². The force obeys Hooke’s law force, where the force is proportional to negative displacement with proportional coefficient called spring constant ³.

An object at position $\vec{r}_i$ attached to a point $\vec{r}_o$ by a spring with natural length $l$ and spring constant $k$ will experience spring force in the form of

$$\tag{1} \vec{F} _{S,i} = -k (r _{io} - l) \hat{r} _{io},$$

where the unit vector is

$$\tag{2} \hat{r} _{io} = \frac{\vec{r} _{io}}{r _{io}},$$

relative position

$$\tag{3} \vec{r} _{io} = \vec{r}_i - \vec{r}_o,$$

and

$$\tag{4} r_{io} = |\vec{r} _{io}| = \sqrt{\vec{r} _{io} \cdot \vec{r} _{io}}$$

is distance to the fixed point.

Eqn (1) can be advanced to two objects connected by a spring, where

$$\tag{5} \vec{F} _{S,ij} = -k (r _{ij} - l) \hat{r} _{ij}$$

is the spring force on object $i$ due to object $j$.

Challenge 1. Show how Eqn (5) can be obtained from Eqn (1) and what is the difference between Eqns (1) and (5).