Work, Energy, and Power

• Principle of Energy

  • Energy cannot be created or destroyed, only transformed from one form to another (Einstein).

  • Kinematics and dynamics focus on change in motion.

  • The concept of energy integral to physics over time.

Energy: An Overview

• Definition of Energy

  • Difficult to provide precise definitions due to various forms.

  • Forms include: gravitational potential, kinetic energy from motion, elastic potential energy (stored in springs), thermal energy (heat), and nuclear energy.

  • Law of Conservation of Energy: In any closed system, energy cannot be created or annihilated; it changes form.

• Work

  • Work is the measure of energy transfer when a force acts over a distance.

  • Unit of Work:

    • Joule (J), where 1 J = 1 N·m.

  • Work Formula:

    • For constant force in the same direction as displacement:[ W = F imes d ]

    • Even though work involves vectors, work itself is scalar. Can be positive, negative, or zero.

Calculating Work

• Example 1:

  • Lifting a 2 kg book, against gravitational force over 3 m:

    • Weight of book = mg = (2 kg)(10 m/s²) = 20 N.

    • Work = 20 N × 3 m = 60 J.

Work at an Angle

• Work when force is applied at an angle:

  • Work Formula:[ W = F imes d imes cos(θ) ]

  • Positive work increases object speed; negative work slows it down.

  • Force perpendicular to motion does zero work.

• Example 2:

  • 15 kg crate pulled at 30° angle with force 69 N: [ W = (F_T imes cos(30°)) imes d = (69 N imes cos(30°)) imes 10 m = 600 J ]

Variable Forces and Work

• Work done by a variable force is calculated by the area under the force vs. displacement graph.

• Kinetic Energy

  • Energy due to motion defined as:[ K = rac{1}{2}mv² ]

  • Work done on an object transferring energy to it, increases kinetic energy.

• Work-Energy Theorem:

  • Relates net work done to changes in kinetic energy:[ W = ΔK = K_{final} - K_{initial} ]

Potential Energy

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

• Gravitational Potential Energy (U):

  • Depends on height above a reference point:[ U = mgh ]

• Examples of Potential Energy:

  • Ball on a table, arrow in a bow.

• Conservative vs Non-Conservative Forces:

  • Work done by conservative forces (like gravity) is path-independent. Non-conservative forces (e.g., friction) are path-dependent.

Conservation of Mechanical Energy

• Total mechanical energy is conserved in absence of non-conservative forces:

  • [ E_{total} = K + U ]

  • Changes in mechanical energy reflect work done: [ K_{initial} + U_{initial} = K_{final} + U_{final} ]

• Example:

  • Falling ball's energy transformation from potential to kinetic.

Power

• Definition of Power:

  • Power is the rate of doing work:[ P = rac{W}{t} ]

  • Unit of power is watt (W), where 1 W = 1 J/s.

  • Example: Moving a crate with a constant force gives a calculable power output based on work done and time taken.

Summary

• Work causes energy changes. Positive work adds energy, negative work removes it.

• Conservation laws essential in analyzing work, energy, and their transformations.

• Help in understanding and solving complex physical systems.