⚙️AP Physics C: Mechanics Unit 3 – Work, Energy, and Power in Physics

Key Concepts and Definitions

• Work defined as the product of force and displacement in the direction of the force
• Energy described as the capacity to do work and exists in various forms (kinetic, potential, thermal, etc.)
• Kinetic energy associated with an object's motion and depends on its mass and velocity
• Potential energy stored energy due to an object's position or configuration (gravitational, elastic, electric)
• Conservative forces (gravity, springs) allow the conversion between kinetic and potential energy without loss
• Non-conservative forces (friction, air resistance) dissipate energy as heat or other forms
• Power defined as the rate at which work is done or energy is transferred over time

Work and Its Mathematical Representation

• Work calculated using the formula $W = \vec{F} \cdot \vec{d}$, where $\vec{F}$ is the force and $\vec{d}$ is the displacement
• Dot product of force and displacement vectors determines the amount of work done
  • If force and displacement are in the same direction, positive work is done
  • If force and displacement are in opposite directions, negative work is done
  • If force and displacement are perpendicular, no work is done
• Work is a scalar quantity measured in joules (J)
• Work-energy theorem states that the net work done on an object equals the change in its kinetic energy
• Work done by a variable force can be calculated using the area under the force-displacement curve

Energy: Types and Transformations

• Kinetic energy calculated using the formula $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity
• Gravitational potential energy near Earth's surface calculated using $PE_g = mgh$, where $h$ is height above a reference point
• Elastic potential energy stored in a spring calculated using $PE_s = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium
• Energy can be transformed from one form to another (kinetic to potential, potential to kinetic, etc.)
• Energy transformations occur in systems such as pendulums, roller coasters, and elastic collisions
• Mechanical energy defined as the sum of kinetic and potential energy in a system
• Internal energy associated with the microscopic motion and interactions of particles within a system

Conservation of Energy Principle

• Energy cannot be created or destroyed, only transferred or transformed between different forms
• In an isolated system, the total energy remains constant over time
• Conservation of mechanical energy applies to systems with only conservative forces
  • In these systems, the sum of kinetic and potential energy remains constant
  • Energy is converted between kinetic and potential forms without loss
• When non-conservative forces are present, mechanical energy is not conserved
  • Energy is dissipated as heat or other forms, reducing the total mechanical energy
• Conservation of energy is a fundamental principle in physics and applies to various systems and processes

Power: Definition and Calculations

• Power defined as the rate at which work is done or energy is transferred over time
• Mathematically, power is calculated using the formula $P = \frac{W}{\Delta t}$, where $W$ is work and $\Delta t$ is the time interval
• Power is a scalar quantity measured in watts (W), where 1 W = 1 J/s
• Instantaneous power calculated as $P = \vec{F} \cdot \vec{v}$, where $\vec{F}$ is the force and $\vec{v}$ is the velocity
• Average power calculated by dividing the total work done or energy transferred by the total time
• Power output of machines and devices (engines, motors, generators) determines their efficiency and performance

Applications in Real-World Systems

• Roller coasters demonstrate energy transformations between kinetic and potential energy
  • At the top of a hill, the coaster has maximum potential energy and minimum kinetic energy
  • As the coaster descends, potential energy is converted to kinetic energy, increasing speed
• Pendulums exhibit periodic energy conversions between kinetic and potential energy
  • At the highest points, the pendulum has maximum potential energy and minimum kinetic energy
  • At the lowest point, the pendulum has maximum kinetic energy and minimum potential energy
• Hydroelectric power plants convert the potential energy of water into electrical energy
  • Water stored in a reservoir has gravitational potential energy
  • As water flows through turbines, its potential energy is converted to kinetic energy, driving generators
• Elastic collisions (bouncing balls, springs) involve the transfer of energy between kinetic and potential forms
  • During compression, kinetic energy is converted to elastic potential energy
  • During expansion, elastic potential energy is converted back to kinetic energy

Problem-Solving Strategies

• Identify the relevant concepts and principles (work, energy conservation, power)
• Determine the given information and the quantity to be calculated
• Choose an appropriate coordinate system and reference point for potential energy
• Apply the relevant equations and mathematical representations
  • Work-energy theorem: $W_{net} = \Delta KE$
  • Conservation of energy: $KE_i + PE_i = KE_f + PE_f$ (for conservative systems)
  • Power calculations: $P = \frac{W}{\Delta t}$ or $P = \vec{F} \cdot \vec{v}$
• Solve for the unknown quantity using algebra and substitution
• Check the units and reasonableness of the answer

Common Misconceptions and FAQs

• Misconception: Work is always positive
  • Reality: Work can be positive, negative, or zero depending on the direction of force and displacement
• Misconception: Energy is a substance or material
  • Reality: Energy is a scalar quantity that represents the capacity to do work, not a physical substance
• Misconception: An object at rest has no energy
  • Reality: An object at rest can have potential energy due to its position or configuration
• FAQ: Can energy be created or destroyed?
  • No, energy cannot be created or destroyed, only transferred or transformed between different forms
• FAQ: Is mechanical energy always conserved?
  • Mechanical energy is conserved only in systems with conservative forces. Non-conservative forces dissipate energy
• FAQ: What is the difference between energy and power?
  • Energy is the capacity to do work, while power is the rate at which work is done or energy is transferred over time