4 Energy and Work

4.1 Energy and Work

Work Done By Constant and Non-constant Force

The physical concept of work can be mathematically described by the scalar product between the force and the displacement vectors,

$$\begin{align*} \Delta W &= \vec{F} \cdot \Delta\vec{r} = F\Delta r \cos\theta. \end{align*}$$

In the limit when $\Delta\vec{r} \to 0$, then the displacement becomes $\mathrm{d}\vec{r}$ and

$$\begin{align*} đW &= \vec{F} \cdot \mathrm{d}\vec{r}, \end{align*}$$

where $\mathrm{d}\vec{r}$ has the same direction of $\vec{v}$. Note that we use $đW$ instead of $\mathrm{d}W$ due to the fact that $W$ is not a state function. Instead, $W$ depends on the path of the movement.

Power Applied by a Force

$$\begin{align*} P^a &= \lim_{\Delta t \to 0} \frac{\Delta W^a}{\Delta t} = \lim_{\Delta t \to 0} \frac{F_x^a \Delta x}{\Delta t} = F_x^a \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = F_x^a v_x. \end{align*}$$

Generally, the instantaneous power of a non-constant force at time $t$ is

$$\begin{align*} P(t) &= \vec{F}(t) \cdot \vec{v}(t). \end{align*}$$

Kinetic Energy and Work-Kinetic Energy Theorem

The kinetic energy $K$ of a point particle of mass $m$ moving with speed $v$ is defined to be the non-negative scalar quantity

$$\begin{align*} K &= \frac{1}{2}mv^2 = \frac{1}{2}m\vec{v} \cdot \vec{v}. \end{align*}$$ $$\begin{align*} W &= \int_{x_i}^{x_f} F_x(x)\mathrm{d}x = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = K_f - K_i. \end{align*}$$

In a general case, we consider the change of kinetic energy over a tiny time interval $\mathrm{d}t$. The changing rate of kinetic energy is given by

$$\begin{align*} \frac{\mathrm{d}K}{\mathrm{d}t} &= \frac{1}{2}m\frac{\mathrm{d}(\vec{v} \cdot \vec{v})}{\mathrm{d}t} \\ &= \frac{1}{2}m 2\vec{v} \cdot \frac{\mathrm{d}\vec{v}}{\mathrm{d}t} \\ &= m\vec{v} \cdot \vec{a} \\ &= \vec{v} \cdot \vec{F} \\ &= \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \cdot \vec{F} \\ &= \frac{\mathrm{d}W}{\mathrm{d}t}. \end{align*}$$

Kinetic Energy in the COM Reference Frame

For a system of particles, the total kinetic energy is the sum of the individual kinetic energies of all particles:

$$\begin{align*} K &= \sum_i K_i = \sum_i \frac{1}{2}m_i|\vec{v}_i|^2, \end{align*}$$

where $m_i$ is the mass and $\vec{v}_i$ is the velocity of the $i$-th particle.

$$\begin{align*} \text{K}_\text{S} &= \frac{1}{2} \sum m_i |\vec{v}'_i + \vec{u}|^2 \\ &= \frac{1}{2} \sum m_i (\vec{v}'_i + \vec{u}) \cdot (\vec{v}'_i + \vec{u}) \\ &= \frac{1}{2} \sum m_i (\vec{v}'_i \cdot \vec{v}'_i + 2\vec{v}'_i \cdot \vec{u} + \vec{u} \cdot \vec{u}) \\ &= \frac{1}{2} \sum m_i |\vec{v}'_i|^2 + \vec{u} \cdot \left( \sum m_i \vec{v}'_i \right) + \frac{1}{2} |\vec{u}|^2 \sum m_i \\ &= \text{K}_\text{COM} + \frac{1}{2} M u^2, \end{align*}$$

4.2 Potential Energy

Conservative Forces and Changes in Potential Energies of a System

Whenever the work done by a force in moving an object from an initial point to a final point is independent of the path, the force is called a conservative force. In other words, the work done by a conservative force $\vec{F}(\vec{r})$ in going around a closed path $L$ is zero:

$$\begin{align*} \oint_L \vec{F}(\vec{r}) \cdot \mathrm{d}\vec{r} &= 0. \end{align*}$$

Consider a system consisting of two objects interacting through a conservative force. Denote $\vec{F}_{2,1}$ to be the force on object 1 due to the interaction with object 2 and

$$\begin{align*} \mathrm{d}\vec{r}_{2,1} &= \mathrm{d}\vec{r}_1 - \mathrm{d}\vec{r}_2. \end{align*}$$

to be the relative displacement of the two objects. The change in internal potential energy of the system is defined to be the negative of the work done by the conservative force when the objects undergo a relative displacement from the initial state A to the final state B along any displacement that changes the initial state A to the final state B,

$$\begin{align*} \Delta U_{sys} &= -W_c = - \int_A^B \vec{F}_{2,1} \cdot \mathrm{d}\vec{r}_{2,1} \end{align*}$$

Our definition of potential energy only holds for conservative forces, because the work done by a conservative force does not depend on the path but only on the initial and final positions.

In one-dimensional case, with the relationship between integral and derivation we can get the conservative force from potential energy,

$$\begin{align*} F_{2,1} &= -\frac{\mathrm{d}U}{\mathrm{d}x}. \end{align*}$$

And in general, it should be written as a gradient,

$$\begin{align*} \vec{F}_{2,1} &= -\nabla U. \end{align*}$$

Potential Energy Raised by Gravitational Force

$$\begin{align*} W &= \int_{\vec{r}_i}^{\vec{r}_f} \vec{F} \cdot \mathrm{d}\vec{r} = -G m_A m_B \int_{\vec{r}_i}^{\vec{r}_f} \frac{\vec{r} \cdot \mathrm{d}\vec{r}}{r^3} \\ &= -\frac{1}{2} G m_A m_B \int_{\vec{r}_i}^{\vec{r}_f} \frac{\mathrm{d}(r^2)}{r^3} = -G m_A m_B \int_{\vec{r}_i}^{\vec{r}_f} \frac{\mathrm{d}r}{r^2} \\ &= G m_A m_B \left( \frac{1}{r} \right) \Bigg|_{r_i}^{r_f} \\ &= -G m_A m_B \left( \frac{1}{r_i} - \frac{1}{r_f} \right). \end{align*}$$ $$\begin{align*} U_g(r) = -G m_A m_B \frac{1}{r} \end{align*}$$

4.3 Conservation of Mechanical Energy

The Principle of Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in an isolated system that is only subject to conservative forces, the mechanical energy is constant,

$$\begin{align*} \Delta E_m = \Delta K_{sys} + \Delta U_{sys} = 0. \end{align*}$$

Change of Mechanical Energy for Closed System with Internal Non-Conservative Forces

$$\begin{align*} W_{nc} = \Delta K + \Delta U = \Delta E_m. \end{align*}$$

Change of Mechanical Energy for a Non-Closed System

$$\begin{equation*} \Delta E_{sys} = W_{ext} + Q. \end{equation*}$$

This is also called the first law of thermodynamics.