Energy Conservation & Work

2024-10-1611:09 Status: Tags:

For any physical system in an environment with time translation [symmetry][Symmetry (physics)], there is a quantity called energy that is unchanging in time.

Energy is a single, scalar value; can come in different forms;

Kinetic Energy

Energy associated with the motion of an object: $E_{kinetic}=E_k=\frac{1}{2}m{\vec{v}}^2$ Where $E_k$ is the kinetic energy, $m$ is the mass, and $v$ is the velocity = $v_x^2+v_y^2+v_z^2$ and is small compared to the speed of light $3.00\times 10^{8}m/s$. The exact formula for kinetic energy as speed nears the speed of light is. $E=[\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}]m{c}^2$

Thermal Energy

Thermal energy is the kinetic energy of the random motions of molecules

Sound Energy

Work

Work is the net energy transferred to an object by some force. $W=\vec{F}\cdot \delta \vec{x}=F_{||}\delta x$ Where $W$ is work, $\vec{F}$ is the net force, $F_{||}$ is the component of $\vec{F}$ in the direction of motion, and $\delta x$ is the distance travelled. Note that $F_x\delta x+F_y\delta y+F_z\delta z$.

Potential Energy

Potential energy is the energy associated with the configuration of a system (locations of its parts)

Gravitational Potential

$$\begin{array}{c}W=F_{||}\cdot \delta x\\ W=mg\cdot \delta x\\ E=mgh\end{array}$$

Where $E_g$ is the gravitational potential energy, $m$ is the mass of the object, $g$ is the gravitational constant (for small masses on Earth at distances one Earth radius away), and $h$ is the height

Elastic Potential

$$\begin{array}{c}W=F_{||}\cdot \delta x\\ W=-\frac{1}{2}kx\cdot x\\ E=-\frac{1}{2}k{x}^2\end{array}$$

Step 2: comes from the average force exerted on the spring; or the sum of smaller infinitesimally small compressions in x (integrate force as a function of compression

$F=-\frac{dU}{dx}$

Where …

Chemical Potential

Nuclear Potential

Elasticity of Collisions