Suppose a spring lying on a horizontal table, displaced from its equilibrium length by an external agent. The external agent is removed, the spring will head back to its equilibrium length. Here, the direction of spring force and displacement will be same.
But according to Hooke's law,
$$\mathbf{F}=-k\Delta\mathbf{x}$$
The minus sign tells us that the force exerted by spring is always opposite to the direction of displacement.
How is this? Please explain the reason for the minus sign.
Thanks.
Best Answer
Imagine a spring which has a force $\vec F_{\rm sy}$ applied on it by you and this produces an extension $\vec x$.
You then have $F_{\rm sy}=k\vec x$
However it is usual to be interested in the force the spring exerts on you $F_{\rm ys}$.
Using Newton's third law $\vec F_{\rm sy}=-\vec F_{\rm ys }$ so $\vec F_{\rm ys}= - k \vec x$.
Introducing a unit vector in the positive x-direction $\hat i$ and let the magnitude of the forces $F_{\rm ys}$ and $F_{\rm sy}$ be $F$.
$\vec F_{\rm ys}= - k \vec x$ becomes $F \hat i = - kx \hat i \rightarrow F=-kx$ in terms of components in the positive x direction.