9.2 Mechanical Energy and Conservation of Energy

Mechanical energy is all about motion and position. It comes in two flavors: kinetic energy (movement) and potential energy (stored). Understanding how these energies change and interact is key to grasping the bigger picture of energy in physics.

Conservation of energy is a fundamental principle in mechanics. It states that energy can't be created or destroyed, only transformed. This idea helps us understand how energy moves between different forms and why the total energy in a closed system stays constant.

Mechanical Energy

Changes in kinetic and potential energy

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• Kinetic energy ($KE$) represents the energy of motion calculated using the formula $KE = \frac{1}{2}mv^2$ where $m$ is mass and $v$ is velocity
  • Example: a moving car has kinetic energy dependent on its mass and speed
• Potential energy ($PE$) is the energy stored due to an object's position or configuration
  • Gravitational potential energy ($PE_g$) is calculated using the formula $PE_g = mgh$ where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height relative to a reference point
    • Example: a book on a shelf has gravitational potential energy relative to the floor
  • Elastic potential energy ($PE_e$) is the energy stored in a compressed or stretched spring calculated using the formula $PE_e = \frac{1}{2}kx^2$ where $k$ is spring constant and $x$ is displacement from equilibrium position
    • Example: a compressed spring in a toy gun has elastic potential energy
• Changes in energy can be calculated using the following formulas:
  • Change in kinetic energy: $\Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)$ where $v_f$ is final velocity and $v_i$ is initial velocity
  • Change in gravitational potential energy: $\Delta PE_g = mg(h_f - h_i)$ where $h_f$ is final height and $h_i$ is initial height
  • Change in elastic potential energy: $\Delta PE_e = \frac{1}{2}k(x_f^2 - x_i^2)$ where $x_f$ is final displacement and $x_i$ is initial displacement

Conservation of Energy

Conservation of energy in mechanics

• The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another (energy transformation), and in a closed system, total energy remains constant
• Energy transformations occur between kinetic energy and potential energy and vice versa
  • Example: a pendulum swings back and forth, continuously converting kinetic energy to gravitational potential energy and back
• In the absence of non-conservative forces (friction, air resistance), mechanical energy is conserved, meaning the sum of kinetic and potential energy remains constant: $KE_i + PE_i = KE_f + PE_f$ where subscripts $i$ and $f$ denote initial and final states, respectively
  • Example: in a frictionless roller coaster, the total mechanical energy remains constant throughout the ride

Work-energy relationships in physics

• Work ($W$) is the product of force and displacement in the direction of the force, calculated using the formula $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ where $\vec{F}$ is force vector, $\vec{d}$ is displacement vector, and $\theta$ is the angle between $\vec{F}$ and $\vec{d}$
• The work-energy theorem states that the net work done on an object equals the change in its kinetic energy: $W_{net} = \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)$
• Work done by conservative forces:
  1. Gravitational force: $W_g = -\Delta PE_g = -mg(h_f - h_i)$
  2. Spring force: $W_s = -\Delta PE_e = -\frac{1}{2}k(x_f^2 - x_i^2)$
• Work done by non-conservative forces, such as friction $W_f = -f_kd$ where $f_k$ is kinetic friction force and $d$ is displacement, reduces the mechanical energy of the system through energy dissipation
• Power ($P$) is the rate at which work is done, calculated using the formula $P = \frac{W}{\Delta t}$ where $W$ is work and $\Delta t$ is time interval
  • Example: a more powerful engine can perform the same amount of work in less time compared to a less powerful one

Energy in Systems and Equilibrium

• A closed system is one where energy can be transferred between different forms within the system, but no energy enters or leaves the system
• Non-conservative forces, such as friction or air resistance, can cause energy to be dissipated from the system, often in the form of heat
• Mechanical equilibrium occurs when the net force acting on a system is zero, and there is no change in the system's kinetic energy or potential energy over time