Chapter 4: Work and Energy

• Work Done by a Constant Force

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  • Work: Scalar product of force and displacement.
  • $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$.
  • Unit: Joule (J) = N m.
  • Dimensions: $[ML^2T^{-2}]$.
  • Work is a scalar quantity.

  • Conditions for Work Done

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    • Force applied.
    • Displacement occurs.
    • Force component along displacement.

  • Cases of Work Done

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    Angle ($\theta$)	$\cos\theta$	Work Done (W)	Description
    $0^\circ$	1	$Fd$ (Maximum positive)	Force and displacement parallel.
    $90^\circ$	0	0	Force and displacement perpendicular.
    $180^\circ$	-1	$-Fd$ (Maximum negative)	Force and displacement anti-parallel.

    Work done by centripetal force is always zero. Work done by gravitational force on satellite is zero.

• Work Done by a Variable Force

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  • If force changes, work is calculated by dividing displacement into small elements.
  • Total work = sum of work done over each element.
  • Graphically: Area under Force-Displacement graph.

• Work Done by Gravitational Field

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  • Work done by gravitational field is independent of path.
  • $W_{gravity} = mgh$.
  • Work done by gravitational field in a closed path is zero.

• Power

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  • Rate of doing work.
  • $P = \frac{W}{t}$.
  • Unit: Watt (W) = J/s.
  • Dimensions: $[ML^2T^{-3}]$.
  • Scalar quantity.
  • Instantaneous Power (linear): $P = \vec{F} \cdot \vec{v} = Fv \cos\theta$.
  • Instantaneous Power (angular): $P = \vec{\tau} \cdot \vec{\omega} = \tau\omega \cos\theta$.
  • 1 Horsepower (hp) = 746 W.

• Energy

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  • Capacity to do work.
  • Unit: Joule (J).
  • Dimensions: $[ML^2T^{-2}]$ (same as work).
  • Scalar quantity.

  • Kinetic Energy (KE)

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    • Energy due to motion.
    • $KE = \frac{1}{2}mv^2$.
    • KE is always positive.
    • Relation with momentum: $KE = \frac{p^2}{2m}$.
    • Percentage change in KE if momentum changes: $\%\Delta KE = (2 + \%\Delta p)\%\Delta p$. (Approximation for small changes)
    • Percentage change in momentum if KE changes: $\%\Delta p = \frac{1}{2}\%\Delta KE$. (Approximation for small changes)

  • Potential Energy (PE)

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    • Energy due to position or configuration.

    Gravitational Potential Energy (GPE)

    • $GPE = mgh$.
    • Depends on reference level.
    • Can be positive, negative, or zero.

    Elastic Potential Energy (EPE)

    • Energy stored in a stretched/compressed spring.
    • $EPE = \frac{1}{2}kx^2$.
    • $k$: spring constant (N/m).
    • $x$: displacement from equilibrium.

• Work-Energy Theorem

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  • Net work done on a body = Change in its kinetic energy.
  • $W_{net} = \Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$.

• Conservation of Energy

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  • Total energy of an isolated system remains constant.
  • Energy cannot be created or destroyed, only transformed.
  • $KE_i + PE_i = KE_f + PE_f$.
  • Total Mechanical Energy ($TME$) = $KE + PE$.
  • $TME$ is conserved in conservative fields (e.g., gravity, elastic).

  • Conservative and Non-Conservative Forces

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    Conservative Force	Non-Conservative Force
    Work done independent of path.	Work done depends on path.
    Work done in closed path is zero.	Work done in closed path is non-zero.
    Associated with potential energy.	Not associated with potential energy.
    e.g., Gravitational, Elastic, Electrostatic.	e.g., Friction, Air drag.

• Efficiency

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  • Ratio of useful output to total input.
  • Efficiency ($\eta$) = $\frac{\text{Output}}{\text{Input}} \times 100\%$.
  • Always less than 100% for real machines.

• Interconversion of Mass and Energy

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  • Einstein's mass-energy equivalence: $E = mc^2$.
  • $c$: speed of light ($3 \times 10^8$ m/s).
  • Small mass conversion yields huge energy.

• Absolute Gravitational Potential Energy

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  • Definition: Work done by gravitational force in moving a body from infinity to a point in the gravitational field.
  • Formula: $U = -\frac{GMm}{r}$.
  • $G$: Universal gravitational constant ($6.67 \times 10^{-11} N m^2 kg^{-2}$).
  • $M$: Mass of the Earth.
  • $m$: Mass of the object.
  • $r$: Distance from the center of the Earth.
  • Always negative (attractive force, zero at infinity).

• Escape Velocity

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  • Definition: Minimum velocity required for an object to escape Earth's gravitational field (or any celestial body's) and never return.
  • Formula: $v_e = \sqrt{\frac{2GM}{R}}$ or $v_e = \sqrt{2gR}$.
  • $G$: Universal gravitational constant.
  • $M$: Mass of the Earth.
  • $R$: Radius of the Earth.
  • $g$: Acceleration due to gravity on Earth's surface.
  • For Earth: $v_e \approx 11.2~km/s$.
  • Independent of the mass or direction of projection of the escaping object.

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