Work, Energy and Simple Machines

• Limitations of Direct Newtonian Analysis:

  • Kinematic equations and Newton's laws of motion describe how applied forces change the motion of objects.

  • When forces change continuously with time or act along complex, irregular paths, applying Newton's laws directly becomes analytically difficult and cumbersome.

  • Formulating motion in terms of work, energy, and power offers a simpler, more powerful framework to analyze interactions and determine initial or final physical states without tracking intermediate trajectories.

• Core Definition of Energy:

  • Energy is fundamentally defined as the capacity to do work.

  • Energy lies at the foundation of all physical interactions, mechanical tasks, and daily activities.

• Daily Sources and Forms of Energy:

  • Biochemical energy: Food provides metabolic chemical energy enabling biological motion, such as walking or manual labor.

  • Electrical energy: Electrical circuits supply energy to operate mechanical actuators, such as rotating an electric ceiling fan.

  • Chemical fuel energy: The combustion of hydrocarbons in fuels provides energy to propel motor vehicles.

Work Done by a Constant Force

• Physical Concept of Mechanical Work:

  • In common language, any physical or mental exertion is loosely termed "work."

  • In scientific terminology, work has a precise mathematical and physical meaning requiring both an applied force and a displacement of the object along the direction of that force.

  • Whenever work is evaluated, both the agency exerting the force and the specific object receiving the action must be explicitly identified.

• Derivation from Lifting Observations:

  • Consider a wheat bag of mass $m = 5\,\text{kg}$ resting on the ground. The downward force due to gravity is $mg$, where $g$ is the acceleration due to gravity.

  • Lifting this bag slowly at constant velocity through a vertical height of $1\,\text{m}$ requires an upward force $F = mg$. The work done on the single bag is designated as $W$.

  • Lifting three identical $5\,\text{kg}$ bags one after another to the same height of $1\,\text{m}$ requires three separate operations of work $W$, resulting in a total work done of $3W$. A lifting machine performing this sequential task consumes three times the fuel.

  • Lifting all three bags together across $1\,\text{m}$ requires an applied force three times larger ($3F = 3mg$) over the same displacement of $1\,\text{m}$. The work performed is identical: $3W$. Applying a larger force over the same distance scales the work done proportionally.

  • Lifting a single $5\,\text{kg}$ bag to a height of $3\,\text{m}$ requires applying the force $F = mg$ over three times the distance, resulting in $3W$ of work. Exerting the same force over a greater distance increases work proportionally.

• Mathematical Formulation for Work:

  • The work done $W$ on an object by a constant force $F$ producing a displacement $s$ parallel to the force is:     $\text{work done} = \text{force applied} \times \text{displacement in the direction of the force}$     $W = F \times s$

  • This relation applies universally across horizontal, vertical, or inclined dimensions as long as the displacement aligns with the line of action of the force.

• Units of Work:

  • The SI unit of force is the newton ($\text{N}$), and the SI unit of displacement is the metre ($\text{m}$).

  • The SI unit of work is the joule ($\text{J}$), named in honor of James Prescott Joule.

  • One joule is defined as the work done when a constant force of $1\,\text{N}$ displaces an object by $1\,\text{m}$ along the direction of the force:     $1\,\text{J} = 1\,\text{N} \times 1\,\text{m}$

  • Expressed in base SI units, since $1\,\text{N} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2}$:     $1\,\text{J} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2} \times 1\,\text{m} = 1\,\text{kg}\,\text{m}^2\,\text{s}^{-2}$

• Scalar Nature and Graphical Interpretation of Work:

  • Force and displacement are vector quantities having magnitude and direction; however, work is a scalar quantity possessing magnitude and sign (positive or negative), but no direction in space.

  • On a graph plotting force along the y-axis against displacement along the x-axis, the work done equals the area bounded by the curve and the displacement axis.

  • For a constant force of $10\,\text{N}$ acting over a displacement of $1\,\text{m}$, the work corresponds to the rectangular area:     $\text{Area} = 10\,\text{N} \times 1\,\text{m} = 10\,\text{J}$

  • If the applied force varies with position, the work done remains equal to the total area under the force-displacement curve between the initial and final positions.

• Conditions for Zero Work Done:

  • Zero Force ($F = 0$): If no net force acts on an object, the work done on it is zero ($W = 0$).

  • Zero Displacement ($s = 0$): If a force is exerted on an immovable body (e.g., pushing firmly against a rigid wall), the displacement is zero, and the scientific work done on the body is strictly zero ($W = 0$).

  • Muscular Exertion vs. Scientific Work: When pushing a rigid wall, a person experiences fatigue because internal muscular fibers repeatedly contract and expand, consuming internal biochemical energy; however, no physical work is transferred to the wall.

  • Perpendicular Force and Displacement: If the applied force acts perpendicular ($90^\circ$) to the displacement vector, no displacement occurs along the line of force, yielding zero work ($W = 0$).

  • Example: A person carrying a heavy box on level ground exerts an upward vertical balancing force equal to the box's weight ($mg$). Because the box displaces horizontally, the vertical displacement is zero, and the vertical holding force performs zero work on the box.

• Sign of Work Done:

  • Positive Work: Occurs when the displacement is in the same direction as the applied force. Example: Pushing a wheelchair forward results in a displacement in the forward direction; the pushing force performs positive work.

  • Negative Work: Occurs when the applied force opposes the direction of displacement. Example: When a goalkeeper catches or blocks an incoming football, the goalkeeper applies a backward-directed retarding force while the ball's displacement continues forward; hence, the goalkeeper performs negative work on the ball.

  • Interactive Work Pairs (Newton's Third Law): In every interaction, action and reaction forces are equal and opposite. When the goalkeeper stops a ball, the goalkeeper does negative work on the ball, while the ball exerts a forward force on the goalkeeper's hands along their backward motion, performing positive work on the goalkeeper.

• Worked Examples on Work Done:

  • Example 7.1 (Lifting and Lowering a Dumbbell):

  • A girl lifts a dumbbell vertically at constant speed and then lowers it down slowly.

  • When lifting up, the upward supporting force matches the weight and points in the same direction as the upward displacement; therefore, she does positive work.

  • When lowering down, her supporting force continues to act upward to prevent free-fall, while the displacement is directed downward. Because force and displacement oppose each other, she does negative work on the dumbbell.

  • Example 7.2 (Stopping a Football):

  • Given: Opposing force exerted by goalkeeper $F = 200\,\text{N}$; displacement of hands while arresting the ball $s = 15\,\text{cm} = 0.15\,\text{m}$ against the direction of the force.

  • Calculation:       $W = F \times (-s)$       $W = 200\,\text{N} \times (-0.15\,\text{m}) = -30\,\text{J}$

  • The goalkeeper performs $-30\,\text{J}$ of work on the ball.

Work-Energy Theorem and Energy Transformations

• The Work-Energy Principle:

  • Performing positive work on an object imparts energy to it, giving it the capacity to perform subsequent work on other bodies.

  • When an object does work against an opposing body, it transfers a corresponding portion of its energy to that body.

  • The work-energy theorem states that the net work done by all external forces acting on an object or system is identically equal to the resulting change in the energy of that object or system:     $\text{work done on an object} = \text{change in its energy}$     $W = \Delta E$

  • This theorem applies universally to single particles, complex collections of bodies, constant forces, and non-constant variable forces.

• Everyday Manifestations of the Work-Energy Theorem:

  • Throwing a cricket ball: A fielder does positive mechanical work accelerating a cricket ball, imparting energy. Upon collision with the stumps, the moving ball exerts a forward force displacing the stumps, doing work on them and transferring its kinetic energy.

  • Raised flowerpot: Work is performed to elevate a flowerpot to a balcony ledge. If it falls, it acquires motion and can exert impact forces, damaging objects below.

  • Example 7.3 (Carrom Board Collisions):

  • A striker is flicked to hit a stationary white coin, which subsequently strikes a stationary black coin.

  • Striker on white coin: The moving striker exerts a forward contact force displacing the white coin in the same direction, doing positive work and increasing the white coin's energy. Simultaneously, by Newton's third law, the white coin exerts an equal and opposite backward force on the striker, doing negative work and decreasing the striker's energy.

  • White coin on black coin: The accelerated white coin exerts a forward force on the black coin along its displacement, doing positive work and transferring kinetic energy to pocket the black coin. The black coin exerts an equal opposing force on the white coin, performing negative work and reducing the white coin's kinetic energy.

• Non-Mechanical Pathways of Energy Transfer:

  • Mechanical work is only one mechanism of transferring energy.

  • Thermal conduction and convection: Energy flows spontaneously across a temperature gradient from an object of higher temperature to one of lower temperature.

  • Thermal radiation: Electromagnetic waves transmit energy across vacuums without requiring a physical medium, such as solar irradiance traveling from the Sun to the Earth.

  • Electrical transmission: Potential differences drive the movement of charges through conductive circuits, delivering energy to loads.

  • Acoustic propagation: Mechanical vibrations oscillate intervening particles, transferring energy through compression and rarefaction waves.

  • Nuclear transformations: Fission and fusion processes, such as nuclear reactions in the core of the Sun, convert mass defects into radiant energy.

• Broad Classification of Energy Forms:

  • Mechanical energy: Energy associated with macroscopic motion or spatial configuration.

  • Thermal energy: Internal kinetic and vibrational energy of atoms and molecules causing warmth.

  • Light energy: Radiant electromagnetic energy enabling visual perception.

  • Sound energy: Coordinated vibrational energy propagated through physical media.

  • Electrical energy: Energy associated with the relative positions and flows of charged particles.

  • Nuclear energy: Latent binding energy stored within atomic nuclei released via fission or fusion.

  • Chemical energy: Potential energy held within chemical bonds between atoms, released during biochemical or combustion reactions.

• Historical Context — James Prescott Joule:

  • James Prescott Joule demonstrated the quantitative equivalence between mechanical energy and thermal energy.

  • His experimental findings confirmed that heat is a form of energy rather than a material fluid (caloric), establishing the principle of conservation of energy across thermodynamics.

Mechanical Energy: Kinetic Energy

• Definition and Characteristics:

  • Mechanical energy comprises both kinetic energy and potential energy.

  • Kinetic energy ($K$) is defined as the energy possessed by an object by virtue of its macroscopic motion.

  • Any stationary object has zero velocity and therefore zero kinetic energy.

  • Kinetic energy is a scalar quantity possessing magnitude only, measured in joules ($\text{J}$).

• Mathematical Derivation of the Kinetic Energy Formula:

  • Consider a body of constant mass $m$ traveling with initial velocity $u$.

  • A constant net force $F$ acts on the body over a linear displacement $s$, accelerating it at rate $a$ to a final velocity $v$.

  • From the third kinematic equation of motion:     $v^2 = u^2 + 2as$     $s = \frac{v^2 - u^2}{2a}$

  • Applying Newton's second law ($F = ma$) into the scientific definition of work ($W = F \times s$):     $W = (ma) \times \left(\frac{v^2 - u^2}{2a}\right)$     $W = \frac{1}{2}m(v^2 - u^2)$     $W = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$

  • By the work-energy theorem, the work done equals the change in kinetic energy ($W = \Delta K$).

  • Setting the initial reference state at rest ($u = 0$), the acquired kinetic energy $K$ is:     $K = \frac{1}{2}mv^2$

• Kinetic Energy Variations:

  • When net positive work is done, velocity increases, and kinetic energy increases ($\Delta K > 0$).

  • When net negative work is done, velocity decreases, and kinetic energy decreases ($\Delta K < 0$).

  • When net work is zero ($W = 0$), velocity remains constant, and kinetic energy does not change ($\Delta K = 0$).

• Worked Examples on Kinetic Energy:

  • Example 7.4 (Effect of Doubling Velocity):

  • Initial state: mass $m$, velocity $v$, kinetic energy $K_1 = \frac{1}{2}mv^2$.

  • If velocity doubles to $2v$:       $K_2 = \frac{1}{2}m(2v)^2 = \frac{1}{2}m(4v^2) = 4 \times \left(\frac{1}{2}mv^2\right) = 4K_1$

  • Quadrupling the velocity increases the kinetic energy by a factor of 4.

  • Example 7.5 (Fast Cricket Delivery):

  • Given: Mass of cricket ball $m = 0.2\,\text{kg}$; delivery speed $v = 154.8\,\text{km\,h}^{-1}$.

  • Convert velocity to SI units:       $v = 154.8 \times \frac{1000\,\text{m}}{3600\,\text{s}} = 43\,\text{m\,s}^{-1}$

  • Kinetic energy calculation:       $K = \frac{1}{2}mv^2 = \frac{1}{2} \times 0.2\,\text{kg} \times (43\,\text{m\,s}^{-1})^2$       $K = 0.1 \times 1849 = 184.9\,\text{J}$

  • Example 7.6 (Arresting a Landing Jet Aircraft):

  • Given: Jet mass $m = 15000\,\text{kg}$; arresting wire constant backward retarding force $F = 367500\,\text{N}$; stopping displacement $s = 100\,\text{m}$; final velocity $v = 0\,\text{m\,s}^{-1}$.

  • Final kinetic energy $K_f = 0\,\text{J}$; initial kinetic energy $K_i = \frac{1}{2} \times 15000\,\text{kg} \times u^2 = 7500 u^2$.

  • Change in kinetic energy:       $\Delta K = K_f - K_i = -7500 u^2$

  • Work done by the arresting cable:       $W = F \times (-s) = 367500\,\text{N} \times (-100\,\text{m}) = -36750000\,\text{J}$

  • Equating through the work-energy theorem:       $-7500 u^2 = -36750000$       $u^2 = \frac{36750000}{7500} = 4900\,\text{m}^2\,\text{s}^{-2}$       $u = \sqrt{4900} = 70\,\text{m\,s}^{-1}$

  • Converting to kilometers per hour:       $u = 70 \times \frac{3600}{1000} = 252\,\text{km\,h}^{-1}$

  • The landing approach velocity of the aircraft was $70\,\text{m\,s}^{-1}$ ($252\,\text{km\,h}^{-1}$).

Mechanical Energy: Potential Energy

• Fundamental Definition:

  • Potential energy is the energy stored within an object as a consequence of shape deformation or within a system of interacting objects as a consequence of their relative spatial configuration.

  • Work performed by an external agent against internal conservative restoring forces is stored as potential energy.

• Mechanisms of Potential Energy Storage:

  • Elastic Deformation:

  • In a slingshot (gulel), stretching the elastic band stores potential energy; upon release, internal restorative forces return the band to its equilibrium shape while accelerating a projectile forward.

  • In archery, pulling the bowstring bends the elastic limbs. The work done in deforming the limbs is stored as elastic potential energy, which converts to arrow kinetic energy upon release.

  • In a mechanical spring, applying an external compressive or tensile force alters its physical length against internal restoring forces. Releasing the spring allows internal forces to do work on an adjacent mass, converting stored elastic potential energy into kinetic energy.

  • Configuration of Interacting Fields:

  • Magnetic interaction: Unlike magnetic poles attract. Separating them requires applying work against mutual magnetic attraction, storing magnetic potential energy. When released, they accelerate toward one another, acquiring kinetic energy capable of crushing small objects placed between them. Conversely, pushing like poles together against magnetic repulsion stores potential energy.

  • Electrostatic interaction: Displacing electric charges against mutual attraction or repulsion stores electrostatic potential energy within the field configuration.

  • Gravitational interaction: Raising a mass above the Earth's surface requires work against mutual gravitational attraction. Upon release, the mass and Earth accelerate toward their mutual center of mass, converting gravitational potential energy into kinetic energy.

• Gravitational Potential Energy:

  • Because Earth is vastly more massive than ordinary terrestrial objects, Earth's displacement during mutual attraction is negligible, allowing gravitational potential energy of the Earth-object system to be attributed entirely to the object.

  • Activity 7.1 (Sand Bed Depressions):

  • A heavy sphere is dropped into a container of loose sand from a height of $1\,\text{m}$, producing a visible crater.

  • Dropping the same sphere from a height of $2\,\text{m}$ into an undisturbed section of sand produces a noticeably deeper depression.

  • Raising the object to a greater elevation requires greater work against gravity, storing more gravitational potential energy. Upon impact, this greater energy performs more mechanical displacement work on the sand granules.

  • Derivation of $U = mgh$:

  • Taking the ground surface as the reference zero-potential level ($U = 0$).

  • To lift a body of mass $m$ vertically at a steady, infinitesimal velocity without acceleration, the applied upward force must balance the downward gravitational force: $F = mg$.

  • For a vertical displacement of height $h$, the work done by the lifting agency is:       $W = F \times h = mg \times h = mgh$

  • Under the work-energy theorem, this work equals the change in potential energy:       $U = mgh$

  • The SI unit of potential energy is the joule ($\text{J}$).

  • Validity restriction: The relation $U = mgh$ assumes a constant acceleration due to gravity $g$, which is valid exclusively in the immediate vicinity of the Earth's surface.

• Worked Example on Gravitational Potential Energy:

  • Example 7.7 (Celebratory Cricket Ball Throw):

  • Given: Ball mass $m = 200\,\text{g} = 0.2\,\text{kg}$; peak height $h = 10\,\text{m}$; take $g = 10\,\text{m\,s}^{-2}$.

  • Calculation of potential energy at apex:       $U = mgh = 0.2\,\text{kg} \times 10\,\text{m\,s}^{-2} \times 10\,\text{m} = 20\,\text{J}$

• Conservative vs. Non-Conservative Work:

  • Work performed against conservative internal forces (gravitational, electrostatic, ideal magnetic, elastic restoring forces) is stored reversibly as potential energy.

  • Work done against non-conservative dissipative forces, such as friction or viscous drag, cannot be stored as reversible mechanical potential energy; it dissipates into non-recoverable thermal and acoustic forms.

Conservation of Mechanical Energy

• Statement of the Law:

  • The total mechanical energy of an isolated system is the sum of its instantaneous kinetic energy and potential energy:     $E = K + U$

  • If only conservative forces perform work on an object, its total mechanical energy remains constant throughout its motion:     $K_1 + U_1 = K_2 + U_2 = \text{constant}$

• Analytical Proof for a Freely Falling Body:

  • Consider an object of mass $m$ held at rest at point A, situated at height $h$ above the ground:

  • Initial velocity $u = 0$.

  • Potential energy: $U_A = mgh$.

  • Kinetic energy: $K_A = \frac{1}{2}m(0)^2 = 0$.

  • Total mechanical energy at point A:       $E_A = K_A + U_A = 0 + mgh = mgh$

  • The object falls freely under gravity for time $t$, reaching point B at an intermediate height $h'$:

  • Downward acceleration is $a = g$.

  • Instantaneous downward velocity:       $v = u + gt = 0 + gt = gt$

  • Vertical distance fallen $s$:       $s = ut + \frac{1}{2}gt^2 = 0 + \frac{1}{2}gt^2 = \frac{1}{2}gt^2$

  • Current height above the ground:       $h' = h - s = h - \frac{1}{2}gt^2$

  • Potential energy at point B:       $U_B = mgh' = mg\left(h - \frac{1}{2}gt^2\right) = mgh - \frac{1}{2}mg^2t^2$

  • Kinetic energy at point B:       $K_B = \frac{1}{2}mv^2 = \frac{1}{2}m(gt)^2 = \frac{1}{2}mg^2t^2$

  • Total