When a mass is attached to a spring, it forms a mass-spring system, where the spring exerts a force in response to an applied force. The spring force, denoted as \( F_s \), acts in the opposite direction of the applied force \( F_A \). This relationship can be expressed mathematically as:

\( F_s = -F_A = -k \cdot x \)

Here, \( k \) represents the spring constant, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the spring force opposes the direction of the applied force. In a static situation where the mass is held in place, the net force is zero, leading to the equation:

\( m \cdot a = 0 \

Since the mass of the spring is negligible, the acceleration \( a \) is also zero when the applied force is balanced by the spring force. However, when the applied force is removed, the spring force becomes the only force acting on the mass, which can be described by:

\( F_s = m \cdot a \

Substituting the spring force equation gives:

\( -k \cdot x = m \cdot a \

This leads to the formula for acceleration:

\( a = -\frac{k}{m} \cdot x \

The negative sign indicates that the acceleration is directed opposite to the displacement. For example, consider a block with a mass of 0.60 kg attached to a spring with a spring constant \( k = 15 \, \text{N/m} \), stretched 0.2 meters from its equilibrium position. To find the spring force acting on the block, we use:

\( F_s = -k \cdot x = -15 \cdot 0.2 = -3 \, \text{N} \

The negative value indicates that the force acts to the left, opposing the stretch. To calculate the acceleration, we apply the derived formula:

\( a = -\frac{15}{0.6} \cdot 0.2 = -5 \, \text{m/s}^2 \

Again, the negative sign signifies that the acceleration is directed to the left, consistent with the direction of the spring force. This understanding of the mass-spring system illustrates the fundamental principles of forces, motion, and equilibrium in physics.