Work and energy

1. Work

Work is defined as the measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement.

• Formula for Work:
  $W = F \times s$
  Where:

  • $W$ = Work done (measured in Joules, $J$)

  • $F$ = Applied force (measured in Newtons, $N$)

  • $s$ = Displacement (measured in meters, $m$)

• Work Done at an Angle:
  $W = F \times s \times \cos(\theta)$
  Where $\theta$ is the angle between the force vector and displacement vector.

• Types of Work:

  • Positive Work: Force and displacement are in the same direction ($\theta = 0^\circ$).

  • Negative Work: Force and displacement are in opposite directions ($\theta = 180^\circ$).

  • Zero Work: Force is perpendicular to displacement ($\theta = 90^\circ$) or displacement is zero ($s = 0$).

• Unit of Work:
  $1\,J = 1\,N \times 1\,m$

2. Energy

Energy is defined as the capacity of a body to do work. It is a scalar quantity.

• SI Unit: Joule ($J$).

2.1 Kinetic Energy ($E_k$)

Energy possessed by an object due to its state of motion.

• Formula: $E_k = \frac{1}{2} m v^2$ Where:

  • $m$ = Mass of the object ($kg$)

  • $v$ = Velocity of the object ($m/s$)

2.2 Potential Energy ($E_p$)

Energy possessed by an object due to its position or shape.

• Formula (Gravitational Potential Energy): $E_p = m g h$ Where:

  • $m$ = Mass of the object ($kg$)

  • $g$ = Acceleration due to gravity ($9.8\,m/s^2$)

  • $h$ = Height above reference level ($m$)

2.3 Law of Conservation of Energy

Energy can neither be created nor destroyed; it can only transform from one form to another. Total mechanical energy remains constant:
$E<em>{total} = E</em>k + E_p = \text{constant}$

3. Power

Power is defined as the rate at which work is done or energy is transformed.

• Formula for Power:
  $P = \frac{W}{t}$
  Where:

  • $P$ = Power (measured in Watts, $W$)

  • $W$ = Work done ($J$)

  • $t$ = Time taken ($s$)

• Unit of Power:
  $1\,W = 1\,J/s$
  Commercial unit of energy: $1\,kWh = 3.6 \times 10^6\,J$

4. Simple Machines

Simple machines are devices that change the magnitude or direction of a force, making work easier.

4.1 Key Quantities

• Load ($L$): Resistance force to be overcome ($N$).

• Effort ($E$): Force applied to the machine ($N$).

• Mechanical Advantage ($MA$): Ratio of load to effort.
  $MA = \frac{L}{E}$

• Velocity Ratio ($VR$): Ratio of distance moved by effort to distance moved by load.
  $VR = \frac{d<em>E}{d</em>L}$

• Efficiency ($\eta$): Ratio of output work to input work.
  $\eta = \frac{MA}{VR} \times 100\%$

4.2 Types of Simple Machines

1. Lever: A rigid bar that rotates around a pivot called a fulcrum.

2. Pulley: A grooved wheel carrying a rope or belt to change direction/magnitude of force.

3. Inclined Plane: A flat surface tilted at an angle to raise loads with reduced effort.

4. Wheel and Axle: A wheel connected to a smaller axle so that force applied to one rotates the other.

5. Wedge: A portable double inclined plane used to split objects.

6. Screw: An inclined plane wrapped around a central cylinder.