Spring forces and Hooke's law Knowledge-Based Study

Study notes

1. The Elastic Restoring Force

   An ideal spring is assumed to be massless and capable of returning to its original unstretched length once deforming forces are removed. When a spring is stretched or compressed by a displacement relative to its equilibrium position, it exerts an internal force that opposes the deformation.

   Robert Hooke observed experimentally that for small deformations, the magnitude of this restoring force is directly proportional to the magnitude of the displacement from equilibrium:

   $$F_s = -k \Delta x$$

   In this equation:

   • $F_s$ represents the spring force (in newtons, N).
   • $k$ is the spring constant (in newtons per meter, N/m), which measures the stiffness of the spring. A larger value of $k$ indicates a stiffer spring.
   • $\Delta x$ is the displacement of the spring's free end from its natural, relaxed equilibrium position (in meters, m).

   Why Does the Negative Sign Matter?

   The negative sign in Hooke's law, $F_s = -k \Delta x$, defines the directional nature of the force. The spring force always acts toward the equilibrium position ($\Delta x = 0$):

   • Tension (Stretching): When the spring is stretched ($\Delta x > 0$), the spring force pulls back inward ($F_s < 0$).
   • Compression (Squeezing): When the spring is compressed ($\Delta x < 0$), the spring force pushes outward ($F_s > 0$).

   By Newton's third law, the force exerted by the spring on an attached object is equal in magnitude and opposite in direction to the force exerted by the object on the spring: $F_{\text{applied}} = -F_s = k \Delta x$.

   Equilibrium and Dynamic Systems

   When solving mechanics problems involving springs, treat the spring force as an external contact force acting on the attached body within a free-body diagram.

   1. Static Equilibrium

   Consider a mass $m$ suspended vertically from a ceiling by an ideal spring. In static equilibrium, the net force is zero:

   $$\Sigma F_y = F_s - mg = 0 \implies k \Delta y = mg$$

   The downward gravitational force is precisely balanced by the upward elastic restoring force, yielding an equilibrium stretch of $\Delta y = \frac{mg}{k}$.

   2. Dynamic Motion

   For a block of mass $m$ moving on a frictionless horizontal surface attached to a spring, Newton's second law along the axis of displacement gives:

   $$\Sigma F_x = ma_x \implies -k x = m a_x$$

   The resulting acceleration is not constant; it is directly proportional to displacement and directed opposite to it: $a_x = -\frac{k}{m} x$.

   Limitations of Hooke's Law

   Hooke's law is a linear empirical model and holds true only within specific boundaries for real materials:

   • Proportional Limit: The maximum stress or displacement up to which the force-displacement relationship remains strictly linear.
   • Elastic Limit: The point beyond which a material undergoes permanent, irreversible deformation (plastic deformation). If deformed past this limit, the spring will not return to its original rest length when the load is removed.

   Within introductory Newtonian mechanics, springs are typically treated as perfectly linear and operating well below their elastic limits.