Work, Energy, and Power

Energy is the universal scalar measure of a dynamical system's state and capacity to perform physical work.

1. Work Done by a Variable Force Field

The mechanical work done by a force vector $\mathbf{F}(\mathbf{r})$ along a spatial curve from $\mathbf{r}_1$ to $\mathbf{r}_2$ is defined by the line integral: $$W_{1\to 2} = \int_{\mathbf{r}_1}^{\mathbf{r}_2} \mathbf{F} \cdot d\mathbf{r} = \int_{x_1}^{x_2} F_x dx + \int_{y_1}^{y_2} F_y dy + \int_{z_1}^{z_2} F_z dz$$ For an ideal linear elastic spring (Hooke's Law $\mathbf{F} = -k x \hat{\mathbf{i}}$): $$W = \int_{x_1}^{x_2} (-kx) dx = -\left[ \frac{1}{2}kx_2^2 - \frac{1}{2}kx_1^2 \right] = -\Delta U_{\text{spring}}$$

2. Instantaneous Mechanical Power

Power is the instantaneous time rate of doing work: $$P = \frac{dW}{dt} = \mathbf{F} \cdot \frac{d\mathbf{r}}{dt} = \mathbf{F} \cdot \mathbf{v}$$ If power is supplied to a vehicle of mass $m$ at a constant rate $P_0$, integrating $m v \frac{dv}{dt} = P_0$ gives $v(t) = \sqrt{\frac{2 P_0 t}{m}}$.

The Work-Energy Theorem is the first integral of Newton's second law with respect to spatial displacement.

1. Analytical Proof for a Single Particle

Consider a particle of constant mass $m$ acted upon by a net force $\mathbf{F}_{\text{net}} = m \frac{d\mathbf{v}}{dt}$. The work done along trajectory $C$ from $t_1$ to $t_2$ is: $$W_{\text{net}} = \int_C \mathbf{F}_{\text{net}} \cdot d\mathbf{r} = \int_{t_1}^{t_2} \left( m \frac{d\mathbf{v}}{dt} \right) \cdot \left( \frac{d\mathbf{r}}{dt} \right) dt = \int_{t_1}^{t_2} m \frac{d\mathbf{v}}{dt} \cdot \mathbf{v} \, dt$$ Using the vector identity $\frac{d}{dt}(v^2) = \frac{d}{dt}(\mathbf{v} \cdot \mathbf{v}) = 2 \mathbf{v} \cdot \frac{d\mathbf{v}}{dt}$: $$W_{\text{net}} = m \int_{t_1}^{t_2} \frac{1}{2} \frac{d(v^2)}{dt} dt = \frac{1}{2}m v_2^2 - \frac{1}{2}m v_1^2 = K_2 - K_1 = \Delta K$$ The Work-Energy Theorem: The total work performed on a particle by all concurrent forces (conservative, non-conservative, and constraint forces) equals the net change in its kinetic energy: $$W_{\text{total}} = \Delta K$$

2. Decomposition into Conservative and Non-Conservative Work

Decomposing forces into conservative $\mathbf{F}_c$ and non-conservative $\mathbf{F}_{nc}$ (e.g., friction, drag): $$W_{\text{total}} = W_c + W_{nc} = -\Delta U + W_{nc} = \Delta K$$ Rearranging gives the general mechanical energy evolution equation: $$\Delta (K + U) = \Delta E_{\text{mech}} = W_{nc}$$ If only conservative forces perform work ($W_{nc} = 0$): $$E_{\text{mech}} = K + U = \text{constant}$$

A force field is conservative if the work it performs along any path depends solely on the initial and final endpoints.

1. Potential Energy Functions in 2D and 3D

For a conservative force field $\mathbf{F}(\mathbf{r})$: $$U(\mathbf{r}) = -\int_{\mathbf{r}_{\text{ref}}}^{\mathbf{r}} \mathbf{F}(\mathbf{r}') \cdot d\mathbf{r}' \iff \mathbf{F}(\mathbf{r}) = -\boldsymbol{\nabla}U(\mathbf{r})$$ In Cartesian components: $$F_x = -\frac{\partial U}{\partial x}, \quad F_y = -\frac{\partial U}{\partial y}, \quad F_z = -\frac{\partial U}{\partial z}$$

2. Equipotential Surfaces and Force Orthogonality

An **equipotential surface** is defined by $U(x, y, z) = C = \text{const}$. Along an infinitesimal displacement $d\mathbf{r}$ tangent to the equipotential surface: $$dU = \boldsymbol{\nabla}U \cdot d\mathbf{r} = 0 \implies -\mathbf{F} \cdot d\mathbf{r} = 0$$ Therefore, conservative force vectors are **strictly perpendicular to equipotential surfaces** everywhere in space, pointing in the direction of steepest descent of potential energy.

For a particle moving in a 1D potential $U(x)$, its motion is completely characterized by the energy conservation relation: $$E = \frac{1}{2}m \dot{x}^2 + U(x) = \text{constant}$$ Solving for velocity $\dot{x} = \frac{dx}{dt}$: $$\frac{dx}{dt} = \pm \sqrt{\frac{2}{m}[E - U(x)]}$$ Because kinetic energy $K = \frac{1}{2}m\dot{x}^2 \ge 0$, motion is physically permitted only where $E \ge U(x)$. Points where $E = U(x)$ are the **turning points**, where velocity vanishes and reverses.

1. Equilibrium Points and Stability Criteria

Equilibrium occurs where the net force vanishes: $$F(x) = -\frac{dU}{dx} = 0$$ The stability of an equilibrium point $x_0$ is dictated by the curvature $\left.\frac{d^2 U}{dx^2}\right|_{x_0}$:

• Stable Equilibrium (Local Minimum): $\left.\frac{d^2 U}{dx^2}\right|_{x_0} > 0$. Displacements produce a restoring force. Small oscillations have angular frequency: $$\omega_0 = \sqrt{\frac{k_{\text{eff}}}{m}}, \quad k_{\text{eff}} = \left.\frac{d^2 U}{dx^2}\right|_{x_0}$$
• Unstable Equilibrium (Local Maximum): $\left.\frac{d^2 U}{dx^2}\right|_{x_0} < 0$. Displacements produce runaway forces away from $x_0$.
• Neutral Equilibrium: $\left.\frac{d^2 U}{dx^2}\right|_{x_0} = 0$.

2. Exact Period of Bounded Oscillations

For a particle trapped between turning points $x_1$ and $x_2$: $$T = 2 \int_{x_1}^{x_2} \frac{dx}{\sqrt{\frac{2}{m}[E - U(x)]}}$$