Various springs acting on a point mass

Question

The force exerted on one spring is $\vec{F}=-k\vec{r}$. Now suppose we have N slinkys with stiffness $k_1,k_2,k_3,...,k_N$ where they have one end tied to fixed points in space with coordinates $R_1,R_2,R_3,...R_N$ and the other end tied on a point mass m. How would I find the expression for the force on the point mass when it is a point with coordinate vector $\vec{r}$. There is also gravity with strength $\vec{g}$

I read somewhere online that I can treat the N slinkys (I'm assuming they're parallel) as one large slinky. Therefore, $k_{eq}=k_1+k_2+...+k_N$. Then $F_{total}=-k_{eq}\vec{r}$ Could I do the same with coordinates? So that the total force is $\vec{F}=-k_{eq}+\vec{g}$. Any help is appreciated.

Answer 1

Assume the point mass to be at position $\vec{r}$. A spring at $\vec{R_i}$ exerts a force along the line of connection. $$\vec{F_i} = -k_i (\vec{r}-\vec{R_i})$$ Summing up and adding a term for gravity, it yields $$ \vec{F} = -\sum_i k_i (\vec{r}-\vec{R_i}) - m\vec{g}$$

I can treat the N slinkys (I'm assuming they're parallel) as one large slinky.

If you're truly considering a point mass, this seems quite impossible for distinct $\vec{R_i}$. If the springs were attached to an extended body however, this works.

You may be aware that the center of mass of any system is affected by the external forces only. More importantly, it does not matter where exactly a force attacks. As long as you're only interested in the COM's motion, you may treat it like it's directly attacking at the center of mass.

Assuming the forces from the springs all acting in parallel on e.g a rigid body, one may replace each $(\vec{r}-\vec{R_i})$ by the same vector, for example $\vec{r}_{com}$. You can see immediatley that the force (on the COM) reduces to $$ \vec{F}_{com} = -\left(\sum_i k_i\right)\vec{r}_{com} $$