Work Power & Energy

1. Work (W)

Work is said to be done when a force causes a displacement in an object.

Definition: \( W = \vec{F} \cdot \vec{s} = F s \cos\theta \)

Where:
– \( W \) = Work done (in Joules, J)
– \( F \) = Magnitude of Force applied (in Newtons, N)
– \( s \) = Magnitude of Displacement (in meters, m)
– \( \theta \) = Angle between the force vector and the displacement vector

Key Cases:

  • Positive Work (\( \theta < 90^\circ \), \( \cos\theta \) is positive): Force promotes displacement. (e.g., pushing a car forward).
  • Zero Work (\( \theta = 90^\circ \), \( \cos90^\circ = 0 \)): Force is perpendicular to displacement. (e.g., gravitational force on a horizontally moving object; centripetal force in circular motion).
  • Negative Work (\( 90^\circ < \theta \leq 180^\circ \), \( \cos\theta \) is negative): Force opposes displacement. (e.g., frictional force, gravity on an upward thrown object).

Work is a Scalar Quantity. It has only magnitude, no direction.

Work done by a Variable Force: \( W = \int \vec{F} \cdot d\vec{s} \) (Area under the Force-Displacement graph).

2. Energy (E)

Energy is the capacity of a body to do work. The SI unit is Joule (J).

Law of Conservation of Energy: Energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy in an isolated system remains constant.

Types of Mechanical Energy:

  • Kinetic Energy (K): Energy possessed by a body due to its motion.\( K = \frac{1}{2} m v^2 \)

    Where \( m \) is mass and \( v \) is velocity.

    Work-Energy Theorem: The net work done by all forces on a body is equal to the change in its kinetic energy.

    \( W_{net} = \Delta K = \frac{1}{2} m v_f^2 – \frac{1}{2} m v_i^2 \)

  • Potential Energy (U): Energy possessed by a body due to its position or configuration.
    • Gravitational P.E.: \( U = m g h \)
      Where \( h \) is height above a reference point.
    • Elastic P.E. (Spring): \( U = \frac{1}{2} k x^2 \)
      Where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

3. Power (P)

Power is the rate of doing work. It is the measure of how fast work is done.

Definition:

\( P = \frac{\text{Work Done}}{\text{Time Taken}} = \frac{W}{t} \)

Instantaneous Power: (When force causes displacement)

\( P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} \)

Where \( \vec{v} \) is the instantaneous velocity.

SI Unit: Watt (W). \( 1 \text{ W} = 1 \text{ J/s} \)

Other Units:
– Horsepower (hp): \( 1 \text{ hp} \approx 746 \text{ W} \)
– Kilowatt-hour (kWh): Commercial unit of energy (not power!). \( 1 \text{ kWh} = 3.6 \times 10^6 \text{ J} \)

4. Important Facts for Competitive Exams

  • The sun is the ultimate source of energy for our planet.
  • In an elastic collision, both momentum and kinetic energy are conserved.
  • In an inelastic collision, momentum is conserved but kinetic energy is not. The lost energy is converted to heat, sound, or deformation.
  • Rocket propulsion works on the principle of conservation of momentum.
  • The work done by the force of gravity depends only on the vertical height difference and not on the path taken. It is an example of a conservative force.
  • Friction is a non-conservative force because the work done by friction depends on the path length.
  • One electron-volt (eV) is the kinetic energy gained by an electron accelerating through a potential difference of 1 volt. \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).
  • The energy of food items is measured in Calories (Cal). \( 1 \text{ Cal} = 4184 \text{ J} \approx 4.2 \text{ kJ} \).

5. Summary of All Formulae & Constants

Formulae:

Concept Formula
Work \( W = F s \cos\theta \)
Kinetic Energy \( K = \frac{1}{2} m v^2 \)
Gravitational Potential Energy \( U_g = m g h \)
Elastic Potential Energy \( U_e = \frac{1}{2} k x^2 \)
Work-Energy Theorem \( W_{net} = \Delta K = K_f – K_i \)
Power (Average) \( P = \frac{W}{t} \)
Power (Instantaneous) \( P = \vec{F} \cdot \vec{v} \)

Constants & Conversions:

Constant/Conversion Value
Acceleration due to gravity (g) \( 9.8 \text{ m/s}^2 \) (approx. \( 10 \text{ m/s}^2 \))
1 Horsepower (hp) 746 Watt (W)
1 Kilowatt-hour (kWh) \( 3.6 \times 10^6 \text{ Joules (J)} \)
1 Calorie (Cal) 4184 Joules (J)
1 Electron-Volt (eV) \( 1.6 \times 10^{-19} \text{ J} \)
  • Work (कार्य) : Work is said to be done when a force is applied and the object moves in the direction of the force.
  • Work, Power & Energy is a scalar quantity.
  • Formula: Work (W)= Force (F)×Displacement (d)×cos⁡(θ)
    • : angle between force and displacement
  • No work is done when displacement is zero (even if force is applied).
  • In uniform circular motion, work is zero because angle is 90°.
  • Units:
    • SI Unit = Joule (J)
      • 1 Joule = ergs
      • 1 calorie = 4.186 J
    • CGS = erg
  • Positive Work: Force and displacement in same direction.
  • Negative Work: Force and displacement in opposite direction.
  • Zero Work: No displacement or force is perpendicular.
  • Power (शक्ति) : Power is the rate at which work is done or energy is transferred.
  • Formula: Power (P)= Power is given by \( P = \frac{W}{t} \), where W is work and t is time.
    • Bigger unit = Kilowatt (kW), Megawatt (MW)
  • 1 kW = 1000 W
  • 1 HP (horsepower) = 746 W 
  • Energy (ऊर्जा) : Energy is the capacity to do work.
  • Work and energy have the same units.
    • CGS = erg
    • Commercial = kilowatt-hour (kWh)
      • Joules
  • Types of Energy:
    • Kinetic Energy (KE): Due to motion
      KE= 
    • PE=mgh

      • Where:
      • m: mass (kg)
      • v: velocity (m/s)
      • g: gravity (9.8 m/s²)
      • h: height (m)
  • Energy Conversion
    • Chemical → Mechanical = engine
    • Electrical → Heat = heater
    • Mechanical → Electrical = Generator
    • Solar → Electrical = Solar panels
    • Potential → Kinetic = Falling object

Work Power & Energy