Mechanical energy Knowledge-Based Study

Study notes

1. Definition of Mechanical Energy

   Mechanical energy is the total energy an object possesses due to its macroscopic motion, its spatial position, or its internal deformation. It represents the physical capacity of a system to perform work and is composed of two primary forms: kinetic energy and potential energy.

   Mathematically, mechanical energy $E_{\text{mech}}$ is defined as the sum of kinetic energy ($E_k$) and potential energy ($E_p$):

   $E_{\text{mech}} = E_k + E_p$

   An object can possess purely kinetic energy, purely potential energy, or a combination of both simultaneously depending on its reference frame, position, and velocity.

   Kinetic and Potential Forms

   The total mechanical energy of an object or system is analyzed through its constituent parts:

   • Kinetic Energy ($E_k$): The energy of an object due to its motion. For an object of mass $m$ moving at speed $v$, it is defined as: $E_k = \frac{1}{2}mv^2$. Because speed is squared, doubling the speed quadruples the kinetic energy.
   • Gravitational Potential Energy ($E_{pg}$): The stored energy due to an object's position in a uniform gravitational field $g$ at height $h$ above a chosen reference level: $E_{pg} = mgh$.
   • Elastic Potential Energy ($E_{pe}$): The energy stored in an elastic medium (such as an ideal spring with spring constant $k$) compressed or stretched by a displacement $x$: $E_{pe} = \frac{1}{2}kx^2$.

   The total potential energy is the sum of all stored conservative energy forms present in the system: $E_p = E_{pg} + E_{pe}$.

   Conservative Systems

   When only conservative forces (such as gravity or ideal spring restoring forces) do work within an isolated system, the total mechanical energy remains constant throughout the motion. This principle is known as the Law of Conservation of Mechanical Energy.

   In such systems, energy transitions continuously between kinetic and potential forms without any loss:

   $E_{\text{mech, initial}} = E_{\text{mech, final}}$

   $E_{k1} + E_{p1} = E_{k2} + E_{p2}$

   For example, when a projectile is dropped from rest in a vacuum:

   • At the maximum height $h$, the speed is zero, meaning all mechanical energy is gravitational potential: $E_{\text{mech}} = mgh$.
   • As the object falls, $h$ decreases while speed increases; potential energy is transformed directly into kinetic energy.
   • Immediately before impact ($h = 0$), the potential energy is zero and the kinetic energy equals the initial mechanical energy: $\frac{1}{2}mv^2 = mgh$.

   Energy Dissipation and Work-Energy Theorem

   In real-world processes, non-conservative forces such as friction, air drag, and material deformation resistance act on moving bodies. Non-conservative forces do work that transforms mechanical energy into non-mechanical forms, predominantly thermal energy and sound.

   The generalized work-energy theorem relates the work done by all non-conservative forces ($W_{\text{nc}}$) to the change in mechanical energy:

   $W_{\text{nc}} = \Delta E_{\text{mech}} = E_{\text{mech, final}} - E_{\text{mech, initial}}$

   Important implications include:

   • When dissipative forces like friction act against the direction of motion, $W_{\text{nc}} < 0$, resulting in a decrease in total mechanical energy.
   • Although mechanical energy decreases in the presence of friction, the total energy of the universe remains constant in accordance with the universal law of conservation of energy.
   • Mechanical energy is strictly conserved only when the net work performed by non-conservative forces is zero ($W_{\text{nc}} = 0$).