Work, Energy, and Power

• Basic Principle:

  • "Energy cannot be created or destroyed; it can only be changed from one form to another." —Albert Einstein

  • Kinematics and dynamics study change in systems.

Energy: An Overview

• Definition of Energy:

  • Energy can be challenging to define precisely.

  • Forms of energy include:

    • Gravitational energy

    • Kinetic energy

    • Potential energy (stored in springs)

    • Thermal energy

    • Nuclear energy

  • Law of Conservation of Energy: Energy cannot be created or destroyed in a closed system; it converts from one form to another.

  • Role of Forces:

    • Forces cause changes in energy; work is the transfer of energy.

Work Defined

• Concept of Work:

  • Work occurs when a force is exerted over a distance (e.g. lifting a book).

• Units of Work:

  • The SI unit of work is the joule (J), defined as

    • 1 joule = 1 newton-meter (N·m).

• Work Calculation:

  • Formula: ( W = F \cdot d ) (for constant force).

    • Work is a scalar quantity, but can be positive, negative, or zero depending on the direction of the force relative to displacement.

Work at an Angle

• Work Calculation with Angles:

  • When force is applied at an angle, the formula adjusts: ( W = Fd \cdot (\cos \theta) ).

  • Example:

    • A crate is pulled at a 30° angle with a tension of 69 N over 10 m:

      • ( W = (69 N \cdot \cos(30°))(10 m) = 600 J. )

Kinetic Energy

• Definition of Kinetic Energy:

  • The energy an object possesses due to its motion: ( K = \frac{1}{2}mv^2 ).

• Transfer of Energy:

  • Work done on the object leads to a change in kinetic energy according to the Work-Energy Theorem: ( W_{total} = \Delta K. )

Potential Energy

• Definition:

  • Energy due to an object's position or configuration.

  • Gravitational Potential Energy: ( U = mgh ).

  • As an object falls, gravitational potential energy converts to kinetic energy.

Conservation of Mechanical Energy

• Principle:

  • In absence of non-conservative forces (friction), mechanical energy conservation holds: ( K_i + U_i = K_f + U_f. )

• Example:

  • A ball released from rest converts all initial potential energy to kinetic energy at the floor.

Power

• Definition of Power:

  • The rate at which work is done: ( P = \frac{W}{t}. )

• Units of Power:

  • Watts (W), where ( 1 W = 1 J/s. )

  • Horsepower conversion: ( 1 hp = 746 W. )

Important Equations

• Work: ( W = Fd \cdot \cos\theta )

• Kinetic Energy: ( K = \frac{1}{2}mv^2 )

• Gravitational Potential Energy: ( U = mgh )

• Conservation of Energy: ( K_i + U_i ± W = K_f + U_f )

• Power: ( P = \frac{W}{t} = Fv )

Summary

• Work is the application of force across a displacement, which changes energy.

• Positive work increases energy in a system; negative work decreases energy.

• Energy is conserved within closed systems.

• Power relates to how fast work is done and is measured in watts.