At which of the following positions does a spring have the least potential energy ?

The correct answer is: The spring is neither compressed nor pulled out and is lying on the ground.

Step-by-Step Explanation:

1. Understanding Potential Energy:

The total potential energy of the spring system consists of two parts:

• Elastic Potential Energy (E_{p, elastic}): Energy stored in a spring due to deformation (compression or stretching).

• Gravitational Potential Energy (E_{p, grav}): Energy possessed by an object due to its position above a reference height (the ground).

2. Formulas:

The elastic potential energy is given by:

$E_{p,\text{elastic}}=\frac{1}{2}k{x}^2$

where $k$ is the spring constant and $x$ is the displacement from its natural (unstretched/uncompressed) length.

The gravitational potential energy relative to the ground is given by:

$E_{p,\text{grav}}=mgh$

where $m$ is the mass, $g$ is acceleration due to gravity, and $h$ is the height above the ground.

3. Evaluating the Options:

• When lying on the ground: Height $h=0$, so gravitational potential energy is minimum ($E_{p,\text{grav}}=0$).

• When neither compressed nor pulled out: Displacement $x=0$, so elastic potential energy is minimum ($E_{p,\text{elastic}}=0$).

Therefore, when the spring is relaxed ($x=0$) and lying on the ground ($h=0$), its total potential energy reaches the absolute minimum value of zero.