Conservation of energy Knowledge-Based Study

Study notes

1. The Fundamental Law

   The Law of Conservation of Energy states that energy cannot be created or destroyed; it can only be transformed from one form into another or transferred from one system to another.

   Key principles include:

   • The total amount of energy in an isolated system remains constant over time.
   • Energy can change its form (such as from chemical to thermal, or gravitational potential to kinetic).
   • Whenever energy appears to be 'lost', it has merely dissipated into forms that are less accessible, typically thermal energy or sound.

   The standard SI unit for all forms of energy and work is the joule (J).

   Conservation of Mechanical Energy

   In many physical scenarios, we focus on mechanical energy ($E_m$), which is the sum of macroscopic kinetic energy ($E_k$) and potential energy ($E_p$):

   $$E_m = E_k + E_p$$

   When only conservative forces (such as gravity or ideal spring forces) do work on an object, mechanical energy is conserved:

   $$E_{k,i} + E_{p,i} = E_{k,f} + E_{p,f}$$

   Key Formulas

   • Kinetic Energy: $E_k = \frac{1}{2}mv^2$, where $m$ is mass in kilograms and $v$ is speed in meters per second.
   • Gravitational Potential Energy: $E_p = mgh$, where $g \approx 9.8\text{ m/s}^2$ (or $9.81\text{ m/s}^2$) and $h$ is height above a chosen reference level in meters.

   Example: An object dropped from rest from a height $h$ converts its entire initial potential energy into kinetic energy just before striking the ground, such that $\frac{1}{2}mv^2 = mgh$, leading to $v = \sqrt{2gh}$.

   Mechanical Energy vs. Total Energy

   In real-world situations, non-conservative forces like friction and air resistance do negative work on moving objects. This work transfers mechanical energy into internal (thermal) energy.

   Work-Energy Relationship

   When non-conservative forces are present, mechanical energy is not conserved, but total energy still is:

   $$E_{m,i} + W_{nc} = E_{m,f}$$

   Here, $W_{nc}$ is the work done by non-conservative forces. Because friction opposes motion, $W_{nc} = -F_f d$, meaning mechanical energy decreases:

   $$E_{m,i} - F_f d = E_{m,f}$$

   The 'lost' mechanical energy is converted into thermal energy, warming up the surfaces and the environment:

   $$\Delta E_{\text{thermal}} = F_f d$$

   How to Apply Conservation of Energy

   1. Define the system: Clearly identify the objects included in your system and whether external forces act on them.
   2. Choose initial and final states: Pick two specific instants (State 1 and State 2) where values are known or needed.
   3. Set a reference height: Choose a convenient baseline where gravitational potential energy is defined as zero ($h = 0$).
   4. Identify all energy forms: Determine if kinetic, gravitational potential, or elastic potential energy is present at each state.
   5. Account for non-conservative work: If friction or drag acts along distance $d$, include $-F_f d$ in the balance equation.
   6. Solve algebraically: Equate initial total energy plus work done to final total energy, and isolate the unknown variable.