02.Work and energy

Work and Energy

Definition of Work

  • Work is said to be done when a body displaces in the direction of the applied force.

  • The formula for work is:[ W = F.d \cos \theta ]

    • Where:

      • (W) = work done

      • (F) = force applied

      • (d) = displacement

      • (\theta) = angle between force and displacement.

Force Components

  • The force can be resolved into components:

    • Parallel to the direction of motion: (F \cos \theta)

    • Perpendicular to the direction of motion: (F \sin \theta)

Simple Pendulum and Circular Motion

  • In a simple pendulum, the force acts at right angles when the pendulum is at the lowest point, requiring centripetal force for circular motion.

  • For circular motion:

    • Work can be expressed as ( w = Fd ) and involves radial (centripetal) and tangential forces.

Mathematical Relations for Work

  1. Basic Work Formula:( W = F imes d )

  2. Using angles:

    • By changing the angle, work done is also affected:

    • ( W = F d \cos \theta )

  3. For gases:

    • Work done on compression or expansion: ( W = P \Delta V )

  4. Energy Relations:

    • Work-energy principle: Work is equal to the change in energy

    • ( ext{Work done} = ext{Change in Kinetic Energy} + ext{Change in Potential Energy} )

    • Example: ( W = (1/2 m v^2){final} - (1/2 m v^2){initial} )

Units of Work

  • SI Unit: Joule (J = Nm)

  • Other Units:

    • Erg

    • Electron-Volt (eV)

    • Kilowatt-Hour (KWh)

Types of Work

  • Positive Work: Force acts in the direction of motion.

  • Negative Work: Force acts in the opposite direction of motion.

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

Graphical Representation

  • The area under a force vs. displacement graph represents work done.

Work Done Examples

  1. Work calculation example:( W = F imes d = (5)(-1) + (2)(3) + (7)(0) )

  2. Given a body of mass 5 kg subjected to a 2N force:

    • Using kinematics to find distance traveled.

Different Energy Types

Kinetic Energy (K.E)

  • ( KE = \frac{1}{2} mv^2 )

Potential Energy (P.E)

  • Gravitational Potential Energy:

    • ( PE = mgh )

  • Elastic Potential Energy:

    • ( PE = \frac{1}{2} k x^2 )

Conservation of Energy

  • Energy can neither be created nor destroyed; it can only change forms.

  • Total mechanical energy (kinetic + potential) remains constant in a closed system.

Power

  • Power is the rate of doing work or consuming energy.

  • Formula: ( P = \frac{W}{t} )

  • SI Unit: Watt (W)

  • Conversion: 1 hp = 746 W

Efficiency

  • Defined as the ratio of useful work output to total work input.

  • Expressed as a percentage:( ext{Efficiency} = \left(\frac{W_{output}}{W_{input}}\right) \times 100 % )