Work, Energy, and Power

• Energy cannot be created or destroyed; it is transformed from one form to another. (Einstein)

• Kinematics and dynamics are centered on change; energy is the measurement of this change.

• The concept of energy emerged in physics over a century after Newton's work.

Energy: An Overview

• Energy is defined through various forces: gravitational, kinetic, thermal, and potential.

• The Law of Conservation of Energy states energy cannot be created or destroyed in a closed system, only transformed.

• Work is the transfer of energy and is defined as the application of force over a distance.

• Work Equation: ( W = F imes d ) where ( F ) is force and ( d ) is distance.

• Unit of work: Joule (J) which is equivalent to Newton-meter (N·m).

Work

• Work is a scalar quantity, though it can be positive, negative, or zero:

  • Positive Work: Increases object speed.

  • Negative Work: Decreases object speed.

  • Zero Work: Force is perpendicular to the direction of motion.

Work Example 1: Lifting a Book

• Lifting a 2 kg book to a height of 3 m:

  • Force exerted (F) = mg = (2 kg)(10 m/s²) = 20 N

  • Work done (W) = F × d = (20 N)(3 m) = 60 J.

Work at an Angle

• For forces applied at an angle, the work done is given by: [ W = F imes d imes \cos(\theta) ]

• Example: A 15 kg crate moved by a rope at a 30° angle with a tension of 69 N over 10 m:

  • [ W = (69 N \times \cos(30°))(10 m) \approx 600 J ]

Work with Variable Forces

• When forces vary, work is determined from the area under a force vs. displacement graph.

Kinetic Energy

• Defined as the energy of an object in motion: ( K = \frac{1}{2}mv^2 )

• Positive work done results in an increase in kinetic energy.

Work–Energy Theorem

• Relates work done to the change in kinetic energy:

  • [ W = K_{final} - K_{initial} ]

• Example: A ball mass 0.10 kg moving at 30 m/s has kinetic energy ( K = \frac{1}{2}(0.10 kg)(30 m/s)^2 = 45 J ).

Potential Energy

• Energy due to position, often gravitational:

  • Gravitational potential energy equation: ( U_g = mgh )

• Example: A 2 kg ball at 1.5 m height has potential energy ( U_g = (2 kg)(10 N/kg)(1.5 m) = 30 J ).

Conservation of Mechanical Energy

• Total mechanical energy (E) remains constant when only conservative forces are acting:

  • [ E = K + U ]

• Example of a falling ball:

  • As the ball falls, potential energy decreases while kinetic energy increases, maintaining total mechanical energy.

Nonconservative Forces

• If work is done by nonconservative forces (like friction), the energy equation adjusts to:

  • [ K_i + U_i + W_{other} = K_f + U_f ]

Power

• Power is the rate of doing work:

  • [ P = \frac{W}{t} ]

• Unit of power: Watt (W) = 1 Joule/second.

• An example involving a mover applying a constant force to move a crate:

  • If 1,800 J of work is done over 20 s, then ( P = \frac{1800 J}{20 s} = 90 W ).

Summary

• Work causes a change in energy: positive work increases, negative work decreases energy.

• Conservation of energy dictates the total initial energy equals final energy.

• Key equations for work and energy:

  • Work: ( W = Fd \cos(\theta) )

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

  • Potential Energy: ( U_g = mgh )

  • For total mechanical energy: ( K_i + U_i = K_f + U_f )

  • Power: ( P = \frac{W}{t} )