Comprehensive Notes on Work, Energy, and Power

Work, Energy, and Power

Introduction

• Review of previous chapters on motion, forces, and gravitation.
• Introduction to the concepts of work, energy, and power.
• Living beings require energy from food for life processes.
• Energy is needed for various activities (playing, reading, thinking, etc.).
• Machines also require energy to function (e.g., fuel for engines).

Work

Definition of Work

• Difference between the everyday use of the term 'work' and its scientific definition.

Examples

• Kamali studying hard expends energy but may not accomplish much 'work' in the scientific sense.
• Pushing a rock that doesn't move is exhausting but involves no work scientifically.
• Standing with a heavy load is tiring but doesn't constitute work in physics if there is no displacement.
• Climbing stairs or a tree involves work according to the scientific definition.
• In day-to-day life, any physical or mental labor is considered work.

Scientific Conception of Work

• Two conditions must be satisfied for work to be done:
  • A force must act on an object.
  • The object must be displaced.
• If either condition is absent, no work is done.
• Examples:
  • Pushing a pebble that moves constitutes work.
  • A girl pulling a trolley that moves constitutes work.
  • Lifting a book involves work.
  • A bullock pulling a cart that moves constitutes work.

Work Done by a Constant Force

• If a constant force $F$ acts on an object, and the object is displaced through a distance $s$ in the direction of the force, then the work done $W$ is:
  $W = F \times s$ (10.1)
• Work has only magnitude and no direction.
• Unit of work: newton-metre (N m) or joule (J).
• 1 J: amount of work done when a force of 1 N displaces an object by 1 m along the line of action of the force.

Positive and Negative Work

• When the force and displacement are in the same direction, the work done is positive.
• When the force and displacement are in opposite directions (retarding force), the work done is negative.
• Example: Lifting an object involves positive work by the applied force and negative work by gravity.

Numerical Examples

• Example 10.1: A force of 5 N displaces an object by 2 m. Work done = $5 N \times 2 m = 10 J$
• Example 10.2: A porter lifts a 15 kg luggage by 1.5 m. Work done = $m \times g \times s = 15 kg \times 10 m/s^2 \times 1.5 m = 225 J$

Energy

Definition of Energy

• Life is impossible without energy, and the demand for it is increasing.
• The Sun is the biggest natural source of energy.
• Energy can also be obtained from nuclei of atoms, Earth's interior, and tides.
• Energy is the capacity to do work.
• An object that does work loses energy, and the object on which work is done gains energy.
• The unit of energy is the same as that of work: joule (J).
• 1 kJ = 1000 J.

Forms of Energy

• Various forms of energy: mechanical (potential + kinetic), heat, chemical, electrical, and light energy.

Kinetic Energy

• Energy possessed by an object due to its motion.
• A moving object can do work.
• Kinetic energy increases with speed.
• Kinetic energy is equal to the work done to make the object acquire that velocity.

Expression for Kinetic Energy

• Consider an object of mass $m$ moving with a uniform velocity $u$.
• A constant force $F$ acts on it, displacing it through a distance $s$.
• Work done: $W = F \times s$
• From the equation of motion: $v^2 – u^2 = 2as$ , therefore $s = \frac{v^2 - u^2}{2a}$
• Force: $F = ma$
• Work done can be written as:
  $W = m \times a \times \frac{v^2 - u^2}{2a} = \frac{1}{2}m(v^2 - u^2)$
• If the object starts from rest (u = 0), then $W = \frac{1}{2}mv^2$
• Kinetic energy: $E_k = \frac{1}{2}mv^2$ (10.5)

Numerical Examples

• Example 10.3: An object of mass 15 kg is moving with a velocity of 4 m/s. Its kinetic energy is:
  $E_k = \frac{1}{2} \times 15 kg \times (4 m/s)^2 = 120 J$
• Example 10.4: Work done to increase the velocity of a 1500 kg car from 30 km/h to 60 km/h:
  • Initial velocity $u = 30 \frac{km}{h} = \frac{25}{3} \frac{m}{s}$
  • Final velocity $v = 60 \frac{km}{h} = \frac{50}{3} \frac{m}{s}$
  • Initial kinetic energy $E_{ki} = \frac{1}{2} \times 1500 kg \times (\frac{25}{3} m/s)^2 = \frac{156250}{3} J$
  • Final kinetic energy $E_{kf} = \frac{1}{2} \times 1500 kg \times (\frac{50}{3} m/s)^2 = \frac{625000}{3} J$
  • Work done = Change in kinetic energy = $E<em>{kf} - E</em>{ki} = 156250 J$

Potential Energy

• Energy stored in an object due to its position or configuration.
• Examples: stretched rubber band, wound-up toy car.

Gravitational Potential Energy

• Energy of an object at a height due to gravity.
• Equal to the work done in raising the object from the ground to that point against gravity.
• Consider an object of mass $m$ raised to a height $h$ from the ground.
• The minimum force required is equal to the weight of the object, $mg$.
• Work done: $W = F \times s = mg \times h = mgh$
• Potential energy: $E_p = mgh$ (10.6)
• The potential energy depends on the chosen zero level.

Numerical Examples

• Example 10.5: The potential energy of a 10 kg object at a height of 6 m is: $E_p = 10 kg \times 9.8 m/s^2 \times 6 m = 588 J$
• Example 10.6: The height of a 12 kg object with a potential energy of 480 J is: $h = \frac{480 J}{12 kg \times 10 m/s^2} = 4 m$

Interconvertibility of Energy Forms

• Energy can be converted from one form to another.

Law of Conservation of Energy

• Energy can only be converted from one form to another; it cannot be created or destroyed.
• The total energy before and after the transformation remains the same.

Example: Free Fall

• An object of mass $m$ falling freely from a height $h$.
• Initially, potential energy = $mgh$, kinetic energy = 0.
• As it falls, potential energy converts into kinetic energy.
• At any point: Potential energy + Kinetic energy = constant.
  $mgh + \frac{1}{2}mv^2 = constant$ (10.7)

Rate of Doing Work (Power)

Definition of Power

• Power is the rate of doing work or the rate of transfer of energy.
• If an agent does work $W$ in time $t$, then power $P = \frac{W}{t}$ (10.8)
• The unit of power is watt (W).
• 1 watt = 1 joule/second ($1 W = 1 J/s$).
• $1 kW = 1000 W = 1000 J/s$
• Average power is the total energy consumed divided by the total time taken.

Numerical Examples

• Example 10.7: Two girls (400 N) climb a rope (8 m). Girl A takes 20 s, and girl B takes 50 s.
  • Power of girl A: $P = \frac{400 N \times 8 m}{20 s} = 160 W$
  • Power of girl B: $P = \frac{400 N \times 8 m}{50 s} = 64 W$
• Example 10.8: A 50 kg boy runs up a staircase of 45 steps in 9 s. Height of each step is 15 cm.
  • Total height $h = 45 \times 0.15 m = 6.75 m$
  • Power: $P = \frac{50 kg \times 10 m/s^2 \times 6.75 m}{9 s} = 375 W$