8.3 Conservation of Energy

Energy conservation is a fundamental principle in physics. It states that energy can't be created or destroyed, only converted between forms. This concept helps us understand and predict the behavior of various systems, from simple pendulums to complex machines.

In mechanics, we focus on kinetic and potential energy. By applying the conservation of energy principle, we can solve problems involving motion, forces, and work. This approach is especially useful when dealing with systems that involve changes in height, speed, or elastic deformation.

Conservation of Energy

Conservation of mechanical energy

• Energy cannot be created or destroyed, only converted from one form to another in a closed system total energy remains constant
• Mechanical energy is the sum of kinetic energy (energy of motion $KE = \frac{1}{2}mv^2$, $m$ is mass and $v$ is velocity) and potential energy (energy stored in a system due to its position or configuration)
  • Gravitational potential energy $PE_g = mgh$, $h$ is the height above a reference level (ball on a shelf)
  • Elastic potential energy $PE_e = \frac{1}{2}kx^2$, $k$ is the spring constant and $x$ is the displacement from equilibrium (compressed spring)
• In the absence of non-conservative forces, mechanical energy is conserved $KE_1 + PE_1 = KE_2 + PE_2$ (pendulum swinging)
• When non-conservative forces are present (friction, air resistance), mechanical energy is not conserved
  • Work done by non-conservative forces ($W_{nc}$) changes the total mechanical energy $KE_1 + PE_1 + W_{nc} = KE_2 + PE_2$ (sliding block)
  • This process often involves energy transformation, where one form of energy is converted into another

Energy calculations in simple systems

• For systems without non-conservative forces, use the conservation of mechanical energy equation to solve for unknown variables
  • A pendulum swinging from a height $h_1$ to a height $h_2$: $mgh_1 + \frac{1}{2}mv_1^2 = mgh_2 + \frac{1}{2}mv_2^2$
• For systems with non-conservative forces, include the work done by these forces in the conservation of energy equation
  • A block sliding down a rough incline: $mgh_1 = \frac{1}{2}mv_2^2 + f_kd$, $f_k$ is the kinetic friction force and $d$ is the distance traveled
• Use the conservation of energy equation to find:
  1. Initial and final velocities (roller coaster)
  2. Heights or displacements (bouncing ball)
  3. Spring constants or other system properties (oscillating spring-mass system)

Conservative vs non-conservative forces

• Conservative forces:
  • Work done by a conservative force is independent of the path taken and depends only on the initial and final positions (gravitational force, spring force, electrostatic force)
• Non-conservative forces:
  • Work done by a non-conservative force depends on the path taken and the distance traveled
  • Non-conservative forces dissipate energy from the system or add energy to the system (friction, air resistance, applied forces like pushing or pulling)
• In the presence of only conservative forces, the total mechanical energy of the system remains constant (Earth-moon system)
• When non-conservative forces are present, the total mechanical energy of the system changes by the amount of work done by the non-conservative forces (damped harmonic oscillator)

Work, Power, and Thermodynamics

• Work is the transfer of energy to a system by a force acting on it
• Power is the rate at which work is done or energy is transferred
• Thermodynamics deals with heat and temperature, and their relation to energy and work
  • The first law of thermodynamics relates to the conservation of energy and states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system