Work and Energy & Conservations
- Elementary Physics 5
Work
Work is defined as the product of the force applied to an object and the distance over which that force is applied, in the direction of the force. The formula for work is given by:
\[ W_{12} = \int_{\b{r}_1}^{\b{r}_2} \b{F} \cdot \dd{\b{s}} \]
It is often denoted as $W$, and its SI unit is the joule ($\mathrm{J=N \cdot m= kg \cdot m^2 / s^2}$).
Power
Power is the rate at which work is done or energy is transferred. It is defined as:
\[ P = \odv{W}{t} \]
Alternatively, power can also be expressed in terms of force and velocity:
\[ \begin{align*} P &= \odv{W}{t} \nl &= \odv{}{t} \int \b{F} \cdot \odv{\b{s}}{t} \dd{t} \nl &= \b{F} \cdot \b{v} \end{align*} \]
It is often denoted as $P$, and its SI unit is the watt ($\mathrm{W=J/s= kg \cdot m^2 / s^3}$).
Kinetic Energy
By the Newton’s second law,
\[ \begin{align*} W_{12} &= \int_{\b{r}_1}^{\b{r}_2} \b{F} \cdot \dd{\b{s}} \nl &= \int_{\b{r}_1}^{\b{r}_2} m \dot{\b{v}} \cdot \odv{\b{s}}{t} \dd{t} \nl &= \int_{t_1}^{t_2} m \dot{\b{v}} \cdot \dd{\b{v}} \nl &= \int_{\b{v}_1}^{\b{v}_2} m \dd{ \left( \frac{1}{2} \abs{\b{v}}^2 \right) } \nl &= \frac{1}{2} m (v_2^2 - v_1^2) \end{align*} \]
The kinetic energy of an object is defined as:
\[ K = \frac{1}{2} m v^2 \]
Then we get:
\[ W = \Delta K \]
This is known as the work-energy theorem. We can conclude that energy is something that can be converted into work, and work is the transfer of energy.
Conservative Forces
A force is said to be conservative if the work done by the force on an object moving from one point to another is independent of the path taken. For conservative forces, the work done is only dependent on the initial and final positions of the object. In other words,
\[ \oint \b{F}_C \cdot \dd{\b{s}} = 0 \]
By the mathematical analysis of the property, a conservative force can be expressed as the negative gradient of some scalar function $U$:
\[ \b{F}_C = -\grad U \]
and $U$ is called the potential energy by the conservative force.
Potential Energy
The potential energy associated with a conservative force is defined as the work done by the force when moving an object from a reference point to a specific point in space. The potential energy is given by:
\[ U(\b{r}) = -\int_{\b{r}_0}^{\b{r}} \b{F}_C \cdot \dd{\b{s}} \]
where $\b{r}_0$ is the reference point and can be chosen arbitrarily. The work done by the conservative force is then:
\[ W = -\Delta U \]
We have some common examples of conservative forces and their associated potential energies:
- The gravitational force $\;\b{F} = -m g \hat{\b{z}} \;$ has the potential energy $\;U = m g z$.
- The elastic force of a spring $\;\b{F} = -k \b{x} \;$ has the potential energy $\;U = \frac{1}{2} k x^2$.
Equilibrium points can be found by setting the gradient of the potential energy to zero:
\[ \grad U = 0 \]
The nature of the equilibrium point can be determined by examining the second derivative (or Hessian matrix in multiple dimensions) of the potential energy. If the point is a local minimum, it is a stable equilibrium; if it is a local maximum, it is an unstable equilibrium. If it is a saddle point, it is called an astable equilibrium.
Mechanical Energy
The mechanical energy of a system is the sum of its kinetic energy and potential energy:
\[ E = K + U \]
Conservation Theorems
Linear Momentum Conservation
The linear momentum of an object is conserved if the net external force acting on the object is zero.
\[ \dot{\b{p}} = \b{F}_{\text{net}} \]
This theorem can be analogized to the conservation of the linear momentum of a system of particles, which is trivial by the Newton’s third law.
Mechanical Energy Conservation
The mechanical energy of a system is conserved if the only forces acting on the system are conservative forces. In this case, the total mechanical energy remains constant:
\[ W = \Delta K = -\Delta U \implies \Delta (K + U) = \Delta E = 0 \]
