Hooke's Law, Work, and Energy Conservation in Classical Mechanics

Work and the Work-Energy Theorem

To understand how a spring can move an object, we must look at work. Work is defined as the product of force and displacement, but because both are vectors, we use the dot product to account for the angle between them: $W = \vec{F}_{net} \cdot \vec{d}$

If a force varies along a path (as a spring force does), we calculate work by integrating the force over the displacement: $W = \int_{x_{i}}^{x_{f}} \vec{F}(\vec{x}) \cdot d\vec{x}$

The Work-Energy Theorem connects this work to the object's motion. It states that the net work done on an object is equal to its change in kinetic energy ($K$), which is the energy of motion: $W = \Delta K = K_{f} - K_{i}$ $W = \frac{1}{2}mv_{f}^{2} - \frac{1}{2}mv_{i}^{2}$

Exam Tip: The sign of the work tells you exactly what is happening to the object's speed. If $W > 0$, the object is speeding up ($\Delta K > 0$). If $W < 0$, the object is slowing down ($\Delta K < 0$).

Spring Potential Energy

When work is done against a conservative force like a spring, that energy isn't lost; it is stored as potential energy ($U$). For a spring, this is called spring potential energy.

A spring at rest ($x=0$) has no potential to move an object. However, once it is stretched or compressed, it stores energy based on the spring constant and the square of the displacement: $U_{spring} = \frac{1}{2}kx^{2}$

The relationship between work and potential energy is defined by: $\Delta U = -W$ This means that the work done by the spring is the negative of the change in its potential energy.

Conservation of Energy

In a system where only conservative forces (like gravity or spring forces) are acting, the total mechanical energy remains constant: $K + U = \text{constant}$ $K_{i} + U_{i} = K_{f} + U_{f}$

This principle allows us to solve complex problems by comparing the starting and ending states without needing to know every detail of the path taken.

Gravitational Potential Energy

Just as a spring stores energy when displaced, an object stores gravitational potential energy when lifted against gravity: $U_{gravity} = mgh$

Worked Example: The Navy Jet Landing

Imagine a Navy jet of mass $M$ lands on an aircraft carrier and snags a cable that acts like a spring with constant $k$. The jet is stopped by the spring over a distance $d$. To find the initial speed $v$ of the jet, we equate the initial kinetic energy to the final spring potential energy:

1. Initial State: The jet has kinetic energy $K_i = \frac{1}{2}mv^2$ and no spring energy.
2. Final State: The jet is stopped ($K_f = 0$) and the spring is stretched to $U_f = \frac{1}{2}kd^2$.
3. Conservation: $\frac{1}{2}mv^2 = \frac{1}{2}kd^2$ $v = \sqrt{\frac{k}{m}}d$

Non-Conservative Forces

In the real world, energy is often "lost" to the environment. Non-conservative forces, such as friction, do work that transforms mechanical energy into other forms, like heat. When friction is present, $K + U$ is no longer constant; instead, the total energy decreases.