🍏Principles of Physics I Unit 7 – Potential Energy & Energy Conservation

Key Concepts

• Potential energy stored energy an object possesses due to its position or configuration
• Kinetic energy energy of motion determined by an object's mass and velocity
• Work done when a force acts on an object causing it to move in the direction of the force
• Conservative forces forces that do work independent of the path taken (gravitational, elastic, electrostatic)
  • Work done by conservative forces can be recovered
• Non-conservative forces forces that depend on the path taken (friction, air resistance)
  • Work done by non-conservative forces cannot be recovered
• Mechanical energy sum of an object's potential and kinetic energy in a closed system
• Isolated system no external forces acting on the system allowing for energy conservation

Types of Potential Energy

• Gravitational potential energy (GPE) energy stored in an object due to its height above a reference point
  • Formula: $GPE = mgh$, where $m$ is mass, $g$ is gravitational acceleration, and $h$ is height
• Elastic potential energy energy stored in a deformed elastic object (spring, rubber band)
  • Formula: $EPE = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium
• Electric potential energy energy stored in a system of charged particles due to their relative positions
  • Increases as charges move apart, decreases as they move closer together
• Chemical potential energy energy stored in chemical bonds of molecules
  • Released or absorbed during chemical reactions (combustion, photosynthesis)
• Nuclear potential energy energy stored in the nucleus of an atom due to strong nuclear force
  • Released during nuclear fission or fusion reactions

Energy Conservation Principle

• Energy cannot be created or destroyed, only converted from one form to another
• In an isolated system, total energy remains constant over time
• Changes in potential energy are balanced by changes in kinetic energy
  • As an object falls, GPE decreases while KE increases, total energy remains constant
• Work-energy theorem states that work done on an object equals the change in its kinetic energy
  • $W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$, where $v_f$ and $v_i$ are final and initial velocities
• Conservation of mechanical energy: $\Delta PE + \Delta KE = 0$ (assuming no non-conservative forces)
• Dissipative forces (friction, air resistance) convert mechanical energy into heat or sound
  • Total energy still conserved, but mechanical energy decreases

Mathematical Formulas

• Gravitational potential energy: $GPE = mgh$
  • $m$ is mass (kg), $g$ is gravitational acceleration (9.8 m/s²), $h$ is height (m)
• Elastic potential energy: $EPE = \frac{1}{2}kx^2$
  • $k$ is spring constant (N/m), $x$ is displacement from equilibrium (m)
• Work done by a constant force: $W = Fd\cos\theta$
  • $F$ is force (N), $d$ is displacement (m), $\theta$ is angle between force and displacement
• Kinetic energy: $KE = \frac{1}{2}mv^2$
  • $m$ is mass (kg), $v$ is velocity (m/s)
• Conservation of mechanical energy: $PE_i + KE_i = PE_f + KE_f$
  • Subscripts $i$ and $f$ denote initial and final states

Real-World Applications

• Roller coasters convert GPE to KE during drops, KE back to GPE during climbs
  • Friction gradually reduces total mechanical energy over time
• Bungee jumping relies on elastic potential energy of the bungee cord to safely stop the jumper
  • Cord stretches, converting jumper's KE into EPE, then rebounds
• Hydroelectric dams store GPE in elevated water, convert it to KE as water flows through turbines
  • Turbines generate electricity by rotating magnets within wire coils
• Archery involves storing EPE in the bow as it's drawn, converting it to KE of the arrow upon release
  • Archer's work in drawing the bow is stored as EPE
• Pendulums swing by converting GPE to KE and back, with a gradual decrease due to friction
  • Used in clocks to regulate time, as the period depends on length, not mass or amplitude

Problem-Solving Strategies

• Identify the types of energy present at the initial and final states (PE, KE)
• Determine whether any non-conservative forces are acting (friction, air resistance)
  • If so, mechanical energy is not conserved, use work-energy theorem
• Choose a convenient reference point for measuring gravitational potential energy
  • Often ground level or lowest point in the system
• Apply the principle of conservation of energy, setting initial energy equal to final energy
  • Include all relevant forms of energy (PE, KE, work done by non-conservative forces)
• Solve the resulting equation for the desired quantity (height, velocity, etc.)
  • May need to use quadratic formula or other algebraic techniques
• Check if the answer is reasonable in terms of units and expected behavior
  • Energy should be in joules (J), velocity in meters per second (m/s), etc.

Common Misconceptions

• "Heavier objects fall faster" - in the absence of air resistance, all objects fall with the same acceleration
  • Galileo demonstrated this by dropping objects from the Leaning Tower of Pisa
• "An object at rest has no energy" - an object at rest may have potential energy due to its position
  • A book on a shelf has gravitational potential energy relative to the floor
• "Mechanical energy is always conserved" - only true in the absence of non-conservative forces
  • Friction, air resistance, and other dissipative forces reduce mechanical energy
• "Work is only done when an object moves" - work can be done even if the object doesn't move
  • Holding a heavy box stationary requires work, as you exert a force to counteract gravity
• "Springs always have elastic potential energy" - springs only have EPE when compressed or stretched
  • A relaxed spring at its equilibrium position has no EPE

Connections to Other Physics Topics

• Newton's laws of motion describe forces and acceleration, which relate to work and energy
  • Work-energy theorem derived from Newton's 2nd law: $F = ma$
• Momentum, the product of mass and velocity, is closely related to kinetic energy
  • Collisions involve transfer of both momentum and kinetic energy
• Thermodynamics deals with heat and thermal energy, often converted from mechanical energy
  • Friction converts kinetic energy into thermal energy, increasing entropy
• Electrostatics involves electric potential energy between charged particles
  • Coulomb's law describes the force and potential energy between point charges
• Quantum mechanics uses potential energy curves to model atomic and molecular systems
  • Schrödinger equation describes behavior of particles in various potential energy configurations