Advanced Physics: The Concepts of Work and Energy

1. Work Done by a Force

1.1 General Definition: The Line Integral

In the most general case, a force $\vec{F}$ can vary in magnitude and direction, and the path of its point of application can be a curve in three-dimensional space. The work done by this force as the particle moves from point A to point B along a specific path $C$ is defined by the line integral:

Here, $d\vec{r} = dx\,\hat{i} + dy\,\hat{j} + dz\,\hat{k}$ is the infinitesimal displacement vector along the path $C$. The dot product $\vec{F} \cdot d\vec{r}$ isolates the component of the force that acts along the direction of displacement, which is the only component that contributes to the work done. If the path is parameterized by a variable, say $t$, from $t_A$ to $t_B$, where $\vec{r}(t)$ is the position vector, then the integral becomes:

1.2 Conservative and Non-Conservative Forces

A crucial distinction is made between two types of forces based on the path dependence of the work they do.

• A force is conservative if the work it does on a particle moving between two points is independent of the path taken. This is equivalent to stating that the work done over any closed loop is zero: $\oint \vec{F}_{c} \cdot d\vec{r} = 0$. Examples include gravity and the electrostatic force. Mathematically, a conservative force can always be expressed as the negative gradient of a scalar potential energy function, $U$.
  $\vec{F}_{c} = -\nabla U$
  This property is mathematically equivalent to the force having zero curl: $\nabla \times \vec{F}_{c} = 0$.
• A force is non-conservative if the work it does depends on the path. Friction and air resistance are classic examples. The work done by these dissipative forces is always negative and depends on the total path length.

2. Kinetic Energy and the Work-Energy Theorem

2.1 The Derivation

The Work-Energy Theorem is a direct consequence of Newton's Second Law. Consider the net work done by the total force $\vec{F}_{net}$ on a particle of mass $m$.

We use the chain rule to transform the integral. Knowing that acceleration is the time derivative of velocity, $\vec{a} = \frac{d\vec{v}}{dt}$, and velocity is the time derivative of position, $\vec{v} = \frac{d\vec{r}}{dt}$ (which implies $d\vec{r} = \vec{v} dt$), we can substitute these into the work integral:

The $dt$ terms cancel, allowing us to change the variable of integration from position (via time) to velocity:

To evaluate this integral, we use the product rule for the dot product: $d(\vec{v} \cdot \vec{v}) = d\vec{v} \cdot \vec{v} + \vec{v} \cdot d\vec{v} = 2(\vec{v} \cdot d\vec{v})$. Therefore, $\vec{v} \cdot d\vec{v} = \frac{1}{2} d(\vec{v} \cdot \vec{v})$. Since $\vec{v} \cdot \vec{v} = v^2$, the integral becomes trivial:

We define the kinetic energy as $K = \frac{1}{2}mv^2$. The theorem thus states that the net work done on an object equals the change in its kinetic energy.

2.2 Relativistic Kinetic Energy

In Special Relativity, the total energy of a particle is $E = \gamma m c^2$, where $m$ is the rest mass and $\gamma = (1 - v^2/c^2)^{-1/2}$ is the Lorentz factor. The rest energy is $E_0 = mc^2$. The kinetic energy is the difference between the total energy and the rest energy.

For small velocities ($v \ll c$), the term $x = v^2/c^2$ is very small. We can use the binomial expansion for $\gamma = (1-x)^{-1/2} \approx 1 + (-\frac{1}{2})(-x) + O(x^2)$, which simplifies to $\gamma \approx 1 + \frac{1}{2}\frac{v^2}{c^2}$. Substituting this into the equation for $K_{rel}$ recovers the classical form:

3. Potential Energy and Conservation of Energy

3.1 Potential Energy Function

For any conservative force $\vec{F}_c$, we can define a scalar potential energy function $U(\vec{r})$ such that the work done by the force is equal to the negative change in potential energy.

Only differences in potential energy are physically meaningful; the zero point of potential energy can be chosen arbitrarily. The force is recovered from the potential via the negative gradient operator, $\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$.

For a one-dimensional system, this simplifies to $F_x(x) = -\frac{dU}{dx}$. Points where $F_x=0$ are equilibrium points. The equilibrium is stable if $\frac{d^2U}{dx^2} > 0$ (a local minimum in $U$) and unstable if $\frac{d^2U}{dx^2} < 0$ (a local maximum in $U$).

3.2 The Law of Conservation of Energy

The net work can be split into work done by conservative forces ($W_c$) and work done by non-conservative forces ($W_{nc}$). From the Work-Energy Theorem:

Substituting $W_c = -\Delta U$, we get:

We define the total mechanical energy as $E_{mech} = K + U$. This yields the most general form of the law of energy conservation:

In an isolated system where only conservative forces do work ($W_{nc} = 0$), the total mechanical energy is conserved: $\Delta E_{mech} = 0$, or $K_i + U_i = K_f + U_f$.

4. Theoretical Examples

4.1 Classical: The Simple Pendulum

Consider a simple pendulum of mass $m$ and length $l$. The kinetic energy is $K = \frac{1}{2}mv^2 = \frac{1}{2}m(l\dot{\theta})^2$. The potential energy, setting $U=0$ at the lowest point, is $U = mgh = mgl(1-\cos\theta)$. If energy is conserved, the total energy $E$ is constant:

Differentiating with respect to time gives the equation of motion:

This demonstrates how the equation of motion can be derived elegantly from energy principles.

4.2 Modern Physics: The Morse Potential

The interaction between two atoms in a diatomic molecule can be approximated by the Morse potential, which accounts for bond stretching and dissociation.

Here, $r$ is the interatomic distance, $r_e$ is the equilibrium bond distance, $D_e$ is the well depth (dissociation energy), and $a$ controls the 'width' of the potential. The force between the atoms is $F(r) = -dU/dr$. For small displacements from equilibrium, a Taylor expansion of $U(r)$ around $r_e$ shows that the potential is approximately quadratic, corresponding to a simple harmonic oscillator, which explains the quantized vibrational energy levels of the molecule.