Work, Energy, and Power

• Energy:

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

  • Understanding energy is crucial to Physics; it permeates all branches of the subject.

Energy: An Overview

• Definition: Energy is not easy to define; it varies based on force types (e.g., gravitational, elastic, thermal).

• Law of Conservation of Energy:

  • Energy can’t appear or disappear in a closed system; it transforms from one form to another.

  • Work is the method of transferring energy across systems.

Work

• Definition of Work:

  • Work (W) is done when a force (F) acts over a distance (d):

    • [ W = Fd ]

  • Work is a scalar quantity, can be positive, negative, or zero.

• Units of Work:

  • Measured in Joules (J), where 1 Joule = 1 Newton-meter (N·m).

Calculating Work

• Example 1:

  • Lifting a 2 kg book at constant velocity over 3 m:

    • Weight of book: [ F = mg = 20 N ]

    • Work done: [ W = Fd = (20 N)(3 m) = 60 J ]

Work at an Angle

• Modified Formula:

  • For forces applied at an angle: [ W = Fd \cos(θ) ]

  • Example of a crate at 30° angle:

    • Work done on the crate requiring the angle component.

Examples of Work

• Example 2:

  • Moving 15 kg crate across a horizontal surface at 30°:

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

• Example 3:

  • Work by normal force is zero as it's perpendicular to displacement.

  • Friction does negative work, calculated as: [ W = -µk FN d = -462 J ]

Kinetic Energy

• Definition: Energy of an object due to motion: [ K = mv^2 ]

• Positive work increases kinetic energy.

Work–Energy Theorem

• Energy transfer occurs through work, connecting kinetic energy changes to work done:

  • [ W = K_f - K_i ]

• Example: Kinetic energy of a 0.1 kg ball at 30 m/s gives: [ K = 0.5 mv^2 ]

Potential Energy

• Definition: Arises from an object's position eg. gravitational potential energy:

  • [ U = mgh ] (height relative to a reference point).

• Example: Lifting mass to height has potential energy transformed into kinetic when falling.

Conservation of Mechanical Energy

• Total mechanical energy remains constant when only conservative forces act:

  • [ K_i + U_i = K_f + U_f ]

• Example: Ball at height 5 m has potential energy that changes to kinetic energy.

Work Done by Non-Conservative Forces

• When dealing with non-conservative forces (e.g., friction):

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

Power

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

• Units: Watts (W), where 1 W = 1 J/s.

• Example: Moving a crate with 300 N force over 6 m in 20 s gives power output of 90 W.

Summary

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

• Energy is conserved in a closed system; total initial equals total final energy.

• The power output defines how quickly work is done, measured in watts.